ackbij1lem16 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ackbij1lem16		Structured version Visualization version GIF version

Theorem ackbij1lem16 9057

Description: Lemma for ackbij1 9060. (Contributed by Stefan O'Rear, 18-Nov-2014.)

**Hypothesis**
Ref	Expression
ackbij.f	⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))

**Assertion**
Ref	Expression
ackbij1lem16	⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘𝐴) = (𝐹‘𝐵) → 𝐴 = 𝐵))

Distinct variable groups: 𝑥,𝐹,𝑦 𝑥,𝐴,𝑦 𝑥,𝐵,𝑦

Proof of Theorem ackbij1lem16 Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 3833	. . . . . . . . 9 ⊢ (𝒫 ω ∩ Fin) ⊆ 𝒫 ω
2	1	sseli 3599	. . . . . . . 8 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ 𝒫 ω)
3	2	elpwid 4170	. . . . . . 7 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ⊆ ω)
4	3	adantr 481	. . . . . 6 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → 𝐴 ⊆ ω)
5	1	sseli 3599	. . . . . . . 8 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ 𝒫 ω)
6	5	elpwid 4170	. . . . . . 7 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ⊆ ω)
7	6	adantl 482	. . . . . 6 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → 𝐵 ⊆ ω)
8	4, 7	unssd 3789	. . . . 5 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ⊆ ω)
9		inss2 3834	. . . . . . 7 ⊢ (𝒫 ω ∩ Fin) ⊆ Fin
10	9	sseli 3599	. . . . . 6 ⊢ (𝐴 ∈ (𝒫 ω ∩ Fin) → 𝐴 ∈ Fin)
11	9	sseli 3599	. . . . . 6 ⊢ (𝐵 ∈ (𝒫 ω ∩ Fin) → 𝐵 ∈ Fin)
12		unfi 8227	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐵 ∈ Fin) → (𝐴 ∪ 𝐵) ∈ Fin)
13	10, 11, 12	syl2an 494	. . . . 5 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ∈ Fin)
14		nnunifi 8211	. . . . 5 ⊢ (((𝐴 ∪ 𝐵) ⊆ ω ∧ (𝐴 ∪ 𝐵) ∈ Fin) → ∪ (𝐴 ∪ 𝐵) ∈ ω)
15	8, 13, 14	syl2anc 693	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ∪ (𝐴 ∪ 𝐵) ∈ ω)
16		peano2 7086	. . . 4 ⊢ (∪ (𝐴 ∪ 𝐵) ∈ ω → suc ∪ (𝐴 ∪ 𝐵) ∈ ω)
17	15, 16	syl 17	. . 3 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → suc ∪ (𝐴 ∪ 𝐵) ∈ ω)
18		ineq2 3808	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝐴 ∩ 𝑎) = (𝐴 ∩ ∅))
19	18	fveq2d 6195	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝐹‘(𝐴 ∩ 𝑎)) = (𝐹‘(𝐴 ∩ ∅)))
20		ineq2 3808	. . . . . . . 8 ⊢ (𝑎 = ∅ → (𝐵 ∩ 𝑎) = (𝐵 ∩ ∅))
21	20	fveq2d 6195	. . . . . . 7 ⊢ (𝑎 = ∅ → (𝐹‘(𝐵 ∩ 𝑎)) = (𝐹‘(𝐵 ∩ ∅)))
22	19, 21	eqeq12d 2637	. . . . . 6 ⊢ (𝑎 = ∅ → ((𝐹‘(𝐴 ∩ 𝑎)) = (𝐹‘(𝐵 ∩ 𝑎)) ↔ (𝐹‘(𝐴 ∩ ∅)) = (𝐹‘(𝐵 ∩ ∅))))
23	18, 20	eqeq12d 2637	. . . . . 6 ⊢ (𝑎 = ∅ → ((𝐴 ∩ 𝑎) = (𝐵 ∩ 𝑎) ↔ (𝐴 ∩ ∅) = (𝐵 ∩ ∅)))
24	22, 23	imbi12d 334	. . . . 5 ⊢ (𝑎 = ∅ → (((𝐹‘(𝐴 ∩ 𝑎)) = (𝐹‘(𝐵 ∩ 𝑎)) → (𝐴 ∩ 𝑎) = (𝐵 ∩ 𝑎)) ↔ ((𝐹‘(𝐴 ∩ ∅)) = (𝐹‘(𝐵 ∩ ∅)) → (𝐴 ∩ ∅) = (𝐵 ∩ ∅))))
25	24	imbi2d 330	. . . 4 ⊢ (𝑎 = ∅ → (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘(𝐴 ∩ 𝑎)) = (𝐹‘(𝐵 ∩ 𝑎)) → (𝐴 ∩ 𝑎) = (𝐵 ∩ 𝑎))) ↔ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘(𝐴 ∩ ∅)) = (𝐹‘(𝐵 ∩ ∅)) → (𝐴 ∩ ∅) = (𝐵 ∩ ∅)))))
26		ineq2 3808	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝐴 ∩ 𝑎) = (𝐴 ∩ 𝑏))
27	26	fveq2d 6195	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝐹‘(𝐴 ∩ 𝑎)) = (𝐹‘(𝐴 ∩ 𝑏)))
28		ineq2 3808	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝐵 ∩ 𝑎) = (𝐵 ∩ 𝑏))
29	28	fveq2d 6195	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝐹‘(𝐵 ∩ 𝑎)) = (𝐹‘(𝐵 ∩ 𝑏)))
30	27, 29	eqeq12d 2637	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝐹‘(𝐴 ∩ 𝑎)) = (𝐹‘(𝐵 ∩ 𝑎)) ↔ (𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏))))
31	26, 28	eqeq12d 2637	. . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝐴 ∩ 𝑎) = (𝐵 ∩ 𝑎) ↔ (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)))
32	30, 31	imbi12d 334	. . . . 5 ⊢ (𝑎 = 𝑏 → (((𝐹‘(𝐴 ∩ 𝑎)) = (𝐹‘(𝐵 ∩ 𝑎)) → (𝐴 ∩ 𝑎) = (𝐵 ∩ 𝑎)) ↔ ((𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏))))
33	32	imbi2d 330	. . . 4 ⊢ (𝑎 = 𝑏 → (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘(𝐴 ∩ 𝑎)) = (𝐹‘(𝐵 ∩ 𝑎)) → (𝐴 ∩ 𝑎) = (𝐵 ∩ 𝑎))) ↔ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)))))
34		ineq2 3808	. . . . . . . 8 ⊢ (𝑎 = suc 𝑏 → (𝐴 ∩ 𝑎) = (𝐴 ∩ suc 𝑏))
35	34	fveq2d 6195	. . . . . . 7 ⊢ (𝑎 = suc 𝑏 → (𝐹‘(𝐴 ∩ 𝑎)) = (𝐹‘(𝐴 ∩ suc 𝑏)))
36		ineq2 3808	. . . . . . . 8 ⊢ (𝑎 = suc 𝑏 → (𝐵 ∩ 𝑎) = (𝐵 ∩ suc 𝑏))
37	36	fveq2d 6195	. . . . . . 7 ⊢ (𝑎 = suc 𝑏 → (𝐹‘(𝐵 ∩ 𝑎)) = (𝐹‘(𝐵 ∩ suc 𝑏)))
38	35, 37	eqeq12d 2637	. . . . . 6 ⊢ (𝑎 = suc 𝑏 → ((𝐹‘(𝐴 ∩ 𝑎)) = (𝐹‘(𝐵 ∩ 𝑎)) ↔ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))))
39	34, 36	eqeq12d 2637	. . . . . 6 ⊢ (𝑎 = suc 𝑏 → ((𝐴 ∩ 𝑎) = (𝐵 ∩ 𝑎) ↔ (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏)))
40	38, 39	imbi12d 334	. . . . 5 ⊢ (𝑎 = suc 𝑏 → (((𝐹‘(𝐴 ∩ 𝑎)) = (𝐹‘(𝐵 ∩ 𝑎)) → (𝐴 ∩ 𝑎) = (𝐵 ∩ 𝑎)) ↔ ((𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏)) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏))))
41	40	imbi2d 330	. . . 4 ⊢ (𝑎 = suc 𝑏 → (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘(𝐴 ∩ 𝑎)) = (𝐹‘(𝐵 ∩ 𝑎)) → (𝐴 ∩ 𝑎) = (𝐵 ∩ 𝑎))) ↔ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏)) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏)))))
42		ineq2 3808	. . . . . . . 8 ⊢ (𝑎 = suc ∪ (𝐴 ∪ 𝐵) → (𝐴 ∩ 𝑎) = (𝐴 ∩ suc ∪ (𝐴 ∪ 𝐵)))
43	42	fveq2d 6195	. . . . . . 7 ⊢ (𝑎 = suc ∪ (𝐴 ∪ 𝐵) → (𝐹‘(𝐴 ∩ 𝑎)) = (𝐹‘(𝐴 ∩ suc ∪ (𝐴 ∪ 𝐵))))
44		ineq2 3808	. . . . . . . 8 ⊢ (𝑎 = suc ∪ (𝐴 ∪ 𝐵) → (𝐵 ∩ 𝑎) = (𝐵 ∩ suc ∪ (𝐴 ∪ 𝐵)))
45	44	fveq2d 6195	. . . . . . 7 ⊢ (𝑎 = suc ∪ (𝐴 ∪ 𝐵) → (𝐹‘(𝐵 ∩ 𝑎)) = (𝐹‘(𝐵 ∩ suc ∪ (𝐴 ∪ 𝐵))))
46	43, 45	eqeq12d 2637	. . . . . 6 ⊢ (𝑎 = suc ∪ (𝐴 ∪ 𝐵) → ((𝐹‘(𝐴 ∩ 𝑎)) = (𝐹‘(𝐵 ∩ 𝑎)) ↔ (𝐹‘(𝐴 ∩ suc ∪ (𝐴 ∪ 𝐵))) = (𝐹‘(𝐵 ∩ suc ∪ (𝐴 ∪ 𝐵)))))
47	42, 44	eqeq12d 2637	. . . . . 6 ⊢ (𝑎 = suc ∪ (𝐴 ∪ 𝐵) → ((𝐴 ∩ 𝑎) = (𝐵 ∩ 𝑎) ↔ (𝐴 ∩ suc ∪ (𝐴 ∪ 𝐵)) = (𝐵 ∩ suc ∪ (𝐴 ∪ 𝐵))))
48	46, 47	imbi12d 334	. . . . 5 ⊢ (𝑎 = suc ∪ (𝐴 ∪ 𝐵) → (((𝐹‘(𝐴 ∩ 𝑎)) = (𝐹‘(𝐵 ∩ 𝑎)) → (𝐴 ∩ 𝑎) = (𝐵 ∩ 𝑎)) ↔ ((𝐹‘(𝐴 ∩ suc ∪ (𝐴 ∪ 𝐵))) = (𝐹‘(𝐵 ∩ suc ∪ (𝐴 ∪ 𝐵))) → (𝐴 ∩ suc ∪ (𝐴 ∪ 𝐵)) = (𝐵 ∩ suc ∪ (𝐴 ∪ 𝐵)))))
49	48	imbi2d 330	. . . 4 ⊢ (𝑎 = suc ∪ (𝐴 ∪ 𝐵) → (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘(𝐴 ∩ 𝑎)) = (𝐹‘(𝐵 ∩ 𝑎)) → (𝐴 ∩ 𝑎) = (𝐵 ∩ 𝑎))) ↔ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘(𝐴 ∩ suc ∪ (𝐴 ∪ 𝐵))) = (𝐹‘(𝐵 ∩ suc ∪ (𝐴 ∪ 𝐵))) → (𝐴 ∩ suc ∪ (𝐴 ∪ 𝐵)) = (𝐵 ∩ suc ∪ (𝐴 ∪ 𝐵))))))
50		in0 3968	. . . . . 6 ⊢ (𝐴 ∩ ∅) = ∅
51		in0 3968	. . . . . 6 ⊢ (𝐵 ∩ ∅) = ∅
52	50, 51	eqtr4i 2647	. . . . 5 ⊢ (𝐴 ∩ ∅) = (𝐵 ∩ ∅)
53	52	2a1i 12	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘(𝐴 ∩ ∅)) = (𝐹‘(𝐵 ∩ ∅)) → (𝐴 ∩ ∅) = (𝐵 ∩ ∅)))
54		simp13 1093	. . . . . . . . . . . . 13 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏)))
55		3simpa 1058	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) → (𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))))
56		ackbij1lem2 9043	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ 𝐴 → (𝐴 ∩ suc 𝑏) = ({𝑏} ∪ (𝐴 ∩ 𝑏)))
57	56	fveq2d 6195	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ 𝐴 → (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘({𝑏} ∪ (𝐴 ∩ 𝑏))))
58	57	3ad2ant2 1083	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) ∧ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘({𝑏} ∪ (𝐴 ∩ 𝑏))))
59		ackbij1lem4 9045	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑏 ∈ ω → {𝑏} ∈ (𝒫 ω ∩ Fin))
60	59	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) → {𝑏} ∈ (𝒫 ω ∩ Fin))
61		simprl 794	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) → 𝐴 ∈ (𝒫 ω ∩ Fin))
62		inss1 3833	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∩ 𝑏) ⊆ 𝐴
63		ackbij.f	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐹 = (𝑥 ∈ (𝒫 ω ∩ Fin) ↦ (card‘∪ 𝑦 ∈ 𝑥 ({𝑦} × 𝒫 𝑦)))
64	63	ackbij1lem11 9052	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝑏) ⊆ 𝐴) → (𝐴 ∩ 𝑏) ∈ (𝒫 ω ∩ Fin))
65	61, 62, 64	sylancl 694	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) → (𝐴 ∩ 𝑏) ∈ (𝒫 ω ∩ Fin))
66		incom 3805	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑏} ∩ (𝐴 ∩ 𝑏)) = ((𝐴 ∩ 𝑏) ∩ {𝑏})
67		inss2 3834	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∩ 𝑏) ⊆ 𝑏
68		nnord 7073	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑏 ∈ ω → Ord 𝑏)
69		orddisj 5762	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (Ord 𝑏 → (𝑏 ∩ {𝑏}) = ∅)
70	68, 69	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑏 ∈ ω → (𝑏 ∩ {𝑏}) = ∅)
71	70	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) → (𝑏 ∩ {𝑏}) = ∅)
72		ssdisj 4026	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∩ 𝑏) ⊆ 𝑏 ∧ (𝑏 ∩ {𝑏}) = ∅) → ((𝐴 ∩ 𝑏) ∩ {𝑏}) = ∅)
73	67, 71, 72	sylancr 695	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) → ((𝐴 ∩ 𝑏) ∩ {𝑏}) = ∅)
74	66, 73	syl5eq 2668	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) → ({𝑏} ∩ (𝐴 ∩ 𝑏)) = ∅)
75	63	ackbij1lem9 9050	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑏} ∈ (𝒫 ω ∩ Fin) ∧ (𝐴 ∩ 𝑏) ∈ (𝒫 ω ∩ Fin) ∧ ({𝑏} ∩ (𝐴 ∩ 𝑏)) = ∅) → (𝐹‘({𝑏} ∪ (𝐴 ∩ 𝑏))) = ((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐴 ∩ 𝑏))))
76	60, 65, 74, 75	syl3anc 1326	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) → (𝐹‘({𝑏} ∪ (𝐴 ∩ 𝑏))) = ((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐴 ∩ 𝑏))))
77	76	3ad2ant1 1082	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) ∧ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘({𝑏} ∪ (𝐴 ∩ 𝑏))) = ((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐴 ∩ 𝑏))))
78	58, 77	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) ∧ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝐴 ∩ suc 𝑏)) = ((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐴 ∩ 𝑏))))
79	55, 78	syl3an1 1359	. . . . . . . . . . . . 13 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝐴 ∩ suc 𝑏)) = ((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐴 ∩ 𝑏))))
80		ackbij1lem2 9043	. . . . . . . . . . . . . . . . 17 ⊢ (𝑏 ∈ 𝐵 → (𝐵 ∩ suc 𝑏) = ({𝑏} ∪ (𝐵 ∩ 𝑏)))
81	80	fveq2d 6195	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 ∈ 𝐵 → (𝐹‘(𝐵 ∩ suc 𝑏)) = (𝐹‘({𝑏} ∪ (𝐵 ∩ 𝑏))))
82	81	3ad2ant3 1084	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) ∧ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝐵 ∩ suc 𝑏)) = (𝐹‘({𝑏} ∪ (𝐵 ∩ 𝑏))))
83		simprr 796	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) → 𝐵 ∈ (𝒫 ω ∩ Fin))
84		inss1 3833	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐵 ∩ 𝑏) ⊆ 𝐵
85	63	ackbij1lem11 9052	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ (𝐵 ∩ 𝑏) ⊆ 𝐵) → (𝐵 ∩ 𝑏) ∈ (𝒫 ω ∩ Fin))
86	83, 84, 85	sylancl 694	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) → (𝐵 ∩ 𝑏) ∈ (𝒫 ω ∩ Fin))
87		incom 3805	. . . . . . . . . . . . . . . . . 18 ⊢ ({𝑏} ∩ (𝐵 ∩ 𝑏)) = ((𝐵 ∩ 𝑏) ∩ {𝑏})
88		inss2 3834	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐵 ∩ 𝑏) ⊆ 𝑏
89		ssdisj 4026	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐵 ∩ 𝑏) ⊆ 𝑏 ∧ (𝑏 ∩ {𝑏}) = ∅) → ((𝐵 ∩ 𝑏) ∩ {𝑏}) = ∅)
90	88, 71, 89	sylancr 695	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) → ((𝐵 ∩ 𝑏) ∩ {𝑏}) = ∅)
91	87, 90	syl5eq 2668	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) → ({𝑏} ∩ (𝐵 ∩ 𝑏)) = ∅)
92	63	ackbij1lem9 9050	. . . . . . . . . . . . . . . . 17 ⊢ (({𝑏} ∈ (𝒫 ω ∩ Fin) ∧ (𝐵 ∩ 𝑏) ∈ (𝒫 ω ∩ Fin) ∧ ({𝑏} ∩ (𝐵 ∩ 𝑏)) = ∅) → (𝐹‘({𝑏} ∪ (𝐵 ∩ 𝑏))) = ((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐵 ∩ 𝑏))))
93	60, 86, 91, 92	syl3anc 1326	. . . . . . . . . . . . . . . 16 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) → (𝐹‘({𝑏} ∪ (𝐵 ∩ 𝑏))) = ((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐵 ∩ 𝑏))))
94	93	3ad2ant1 1082	. . . . . . . . . . . . . . 15 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) ∧ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘({𝑏} ∪ (𝐵 ∩ 𝑏))) = ((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐵 ∩ 𝑏))))
95	82, 94	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) ∧ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝐵 ∩ suc 𝑏)) = ((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐵 ∩ 𝑏))))
96	55, 95	syl3an1 1359	. . . . . . . . . . . . 13 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝐵 ∩ suc 𝑏)) = ((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐵 ∩ 𝑏))))
97	54, 79, 96	3eqtr3d 2664	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐴 ∩ 𝑏))) = ((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐵 ∩ 𝑏))))
98	63	ackbij1lem10 9051	. . . . . . . . . . . . . . . . 17 ⊢ 𝐹:(𝒫 ω ∩ Fin)⟶ω
99	98	ffvelrni 6358	. . . . . . . . . . . . . . . 16 ⊢ ({𝑏} ∈ (𝒫 ω ∩ Fin) → (𝐹‘{𝑏}) ∈ ω)
100	60, 99	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) → (𝐹‘{𝑏}) ∈ ω)
101	98	ffvelrni 6358	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∩ 𝑏) ∈ (𝒫 ω ∩ Fin) → (𝐹‘(𝐴 ∩ 𝑏)) ∈ ω)
102	65, 101	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) → (𝐹‘(𝐴 ∩ 𝑏)) ∈ ω)
103	98	ffvelrni 6358	. . . . . . . . . . . . . . . 16 ⊢ ((𝐵 ∩ 𝑏) ∈ (𝒫 ω ∩ Fin) → (𝐹‘(𝐵 ∩ 𝑏)) ∈ ω)
104	86, 103	syl 17	. . . . . . . . . . . . . . 15 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) → (𝐹‘(𝐵 ∩ 𝑏)) ∈ ω)
105		nnacan 7708	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘{𝑏}) ∈ ω ∧ (𝐹‘(𝐴 ∩ 𝑏)) ∈ ω ∧ (𝐹‘(𝐵 ∩ 𝑏)) ∈ ω) → (((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐴 ∩ 𝑏))) = ((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐵 ∩ 𝑏))) ↔ (𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏))))
106	100, 102, 104, 105	syl3anc 1326	. . . . . . . . . . . . . 14 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin))) → (((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐴 ∩ 𝑏))) = ((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐵 ∩ 𝑏))) ↔ (𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏))))
107	106	3adant3 1081	. . . . . . . . . . . . 13 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) → (((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐴 ∩ 𝑏))) = ((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐵 ∩ 𝑏))) ↔ (𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏))))
108	107	3ad2ant1 1082	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐴 ∩ 𝑏))) = ((𝐹‘{𝑏}) +_𝑜 (𝐹‘(𝐵 ∩ 𝑏))) ↔ (𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏))))
109	97, 108	mpbid 222	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)))
110		uneq2 3761	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏) → ({𝑏} ∪ (𝐴 ∩ 𝑏)) = ({𝑏} ∪ (𝐵 ∩ 𝑏)))
111	110	adantl 482	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)) → ({𝑏} ∪ (𝐴 ∩ 𝑏)) = ({𝑏} ∪ (𝐵 ∩ 𝑏)))
112	56	ad2antrr 762	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)) → (𝐴 ∩ suc 𝑏) = ({𝑏} ∪ (𝐴 ∩ 𝑏)))
113	80	ad2antlr 763	. . . . . . . . . . . . . 14 ⊢ (((𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)) → (𝐵 ∩ suc 𝑏) = ({𝑏} ∪ (𝐵 ∩ 𝑏)))
114	111, 112, 113	3eqtr4d 2666	. . . . . . . . . . . . 13 ⊢ (((𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) ∧ (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏))
115	114	ex 450	. . . . . . . . . . . 12 ⊢ ((𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏)))
116	115	3adant1 1079	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ((𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏)))
117	109, 116	embantd 59	. . . . . . . . . 10 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (((𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏)))
118	117	3exp 1264	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) → (𝑏 ∈ 𝐴 → (𝑏 ∈ 𝐵 → (((𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏)))))
119		simp13 1093	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ ¬ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏)))
120	119	eqcomd 2628	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ ¬ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (𝐹‘(𝐵 ∩ suc 𝑏)) = (𝐹‘(𝐴 ∩ suc 𝑏)))
121		simp12r 1175	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ ¬ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐵 ∈ (𝒫 ω ∩ Fin))
122		simp12l 1174	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ ¬ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝐴 ∈ (𝒫 ω ∩ Fin))
123		simp11 1091	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ ¬ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ ω)
124		simp3 1063	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ ¬ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐵)
125		simp2 1062	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ ¬ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ¬ 𝑏 ∈ 𝐴)
126	63	ackbij1lem15 9056	. . . . . . . . . . . 12 ⊢ (((𝐵 ∈ (𝒫 ω ∩ Fin) ∧ 𝐴 ∈ (𝒫 ω ∩ Fin)) ∧ (𝑏 ∈ ω ∧ 𝑏 ∈ 𝐵 ∧ ¬ 𝑏 ∈ 𝐴)) → ¬ (𝐹‘(𝐵 ∩ suc 𝑏)) = (𝐹‘(𝐴 ∩ suc 𝑏)))
127	121, 122, 123, 124, 125, 126	syl23anc 1333	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ ¬ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → ¬ (𝐹‘(𝐵 ∩ suc 𝑏)) = (𝐹‘(𝐴 ∩ suc 𝑏)))
128	120, 127	pm2.21dd 186	. . . . . . . . . 10 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ ¬ 𝑏 ∈ 𝐴 ∧ 𝑏 ∈ 𝐵) → (((𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏)))
129	128	3exp 1264	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) → (¬ 𝑏 ∈ 𝐴 → (𝑏 ∈ 𝐵 → (((𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏)))))
130	118, 129	pm2.61d 170	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) → (𝑏 ∈ 𝐵 → (((𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏))))
131		simp13 1093	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏)))
132		simp12l 1174	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → 𝐴 ∈ (𝒫 ω ∩ Fin))
133		simp12r 1175	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → 𝐵 ∈ (𝒫 ω ∩ Fin))
134		simp11 1091	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → 𝑏 ∈ ω)
135		simp2 1062	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → 𝑏 ∈ 𝐴)
136		simp3 1063	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → ¬ 𝑏 ∈ 𝐵)
137	63	ackbij1lem15 9056	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝑏 ∈ ω ∧ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵)) → ¬ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏)))
138	132, 133, 134, 135, 136, 137	syl23anc 1333	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → ¬ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏)))
139	131, 138	pm2.21dd 186	. . . . . . . . . 10 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → (((𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏)))
140	139	3exp 1264	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) → (𝑏 ∈ 𝐴 → (¬ 𝑏 ∈ 𝐵 → (((𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏)))))
141		simp13 1093	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ ¬ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏)))
142		ackbij1lem1 9042	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑏 ∈ 𝐴 → (𝐴 ∩ suc 𝑏) = (𝐴 ∩ 𝑏))
143	142	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → (𝐴 ∩ suc 𝑏) = (𝐴 ∩ 𝑏))
144	143	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐴 ∩ 𝑏)))
145		ackbij1lem1 9042	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑏 ∈ 𝐵 → (𝐵 ∩ suc 𝑏) = (𝐵 ∩ 𝑏))
146	145	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((¬ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → (𝐵 ∩ suc 𝑏) = (𝐵 ∩ 𝑏))
147	146	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ ((¬ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → (𝐹‘(𝐵 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)))
148	144, 147	eqeq12d 2637	. . . . . . . . . . . . . 14 ⊢ ((¬ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → ((𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏)) ↔ (𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏))))
149	148	biimpd 219	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → ((𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏)) → (𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏))))
150	149	3adant1 1079	. . . . . . . . . . . 12 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ ¬ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → ((𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏)) → (𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏))))
151	141, 150	mpd 15	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ ¬ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → (𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)))
152	143, 146	eqeq12d 2637	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → ((𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏) ↔ (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)))
153	152	biimprd 238	. . . . . . . . . . . 12 ⊢ ((¬ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → ((𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏)))
154	153	3adant1 1079	. . . . . . . . . . 11 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ ¬ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → ((𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏)))
155	151, 154	embantd 59	. . . . . . . . . 10 ⊢ (((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) ∧ ¬ 𝑏 ∈ 𝐴 ∧ ¬ 𝑏 ∈ 𝐵) → (((𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏)))
156	155	3exp 1264	. . . . . . . . 9 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) → (¬ 𝑏 ∈ 𝐴 → (¬ 𝑏 ∈ 𝐵 → (((𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏)))))
157	140, 156	pm2.61d 170	. . . . . . . 8 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) → (¬ 𝑏 ∈ 𝐵 → (((𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏))))
158	130, 157	pm2.61d 170	. . . . . . 7 ⊢ ((𝑏 ∈ ω ∧ (𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) ∧ (𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏))) → (((𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏)))
159	158	3exp 1264	. . . . . 6 ⊢ (𝑏 ∈ ω → ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏)) → (((𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏)))))
160	159	com34 91	. . . . 5 ⊢ (𝑏 ∈ ω → ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (((𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏)) → ((𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏)) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏)))))
161	160	a2d 29	. . . 4 ⊢ (𝑏 ∈ ω → (((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘(𝐴 ∩ 𝑏)) = (𝐹‘(𝐵 ∩ 𝑏)) → (𝐴 ∩ 𝑏) = (𝐵 ∩ 𝑏))) → ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘(𝐴 ∩ suc 𝑏)) = (𝐹‘(𝐵 ∩ suc 𝑏)) → (𝐴 ∩ suc 𝑏) = (𝐵 ∩ suc 𝑏)))))
162	25, 33, 41, 49, 53, 161	finds 7092	. . 3 ⊢ (suc ∪ (𝐴 ∪ 𝐵) ∈ ω → ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘(𝐴 ∩ suc ∪ (𝐴 ∪ 𝐵))) = (𝐹‘(𝐵 ∩ suc ∪ (𝐴 ∪ 𝐵))) → (𝐴 ∩ suc ∪ (𝐴 ∪ 𝐵)) = (𝐵 ∩ suc ∪ (𝐴 ∪ 𝐵)))))
163	17, 162	mpcom 38	. 2 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘(𝐴 ∩ suc ∪ (𝐴 ∪ 𝐵))) = (𝐹‘(𝐵 ∩ suc ∪ (𝐴 ∪ 𝐵))) → (𝐴 ∩ suc ∪ (𝐴 ∪ 𝐵)) = (𝐵 ∩ suc ∪ (𝐴 ∪ 𝐵))))
164		omsson 7069	. . . . . . . 8 ⊢ ω ⊆ On
165	8, 164	syl6ss 3615	. . . . . . 7 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ⊆ On)
166		onsucuni 7028	. . . . . . 7 ⊢ ((𝐴 ∪ 𝐵) ⊆ On → (𝐴 ∪ 𝐵) ⊆ suc ∪ (𝐴 ∪ 𝐵))
167	165, 166	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∪ 𝐵) ⊆ suc ∪ (𝐴 ∪ 𝐵))
168	167	unssad 3790	. . . . 5 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → 𝐴 ⊆ suc ∪ (𝐴 ∪ 𝐵))
169		df-ss 3588	. . . . 5 ⊢ (𝐴 ⊆ suc ∪ (𝐴 ∪ 𝐵) ↔ (𝐴 ∩ suc ∪ (𝐴 ∪ 𝐵)) = 𝐴)
170	168, 169	sylib 208	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐴 ∩ suc ∪ (𝐴 ∪ 𝐵)) = 𝐴)
171	170	fveq2d 6195	. . 3 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐹‘(𝐴 ∩ suc ∪ (𝐴 ∪ 𝐵))) = (𝐹‘𝐴))
172	167	unssbd 3791	. . . . 5 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → 𝐵 ⊆ suc ∪ (𝐴 ∪ 𝐵))
173		df-ss 3588	. . . . 5 ⊢ (𝐵 ⊆ suc ∪ (𝐴 ∪ 𝐵) ↔ (𝐵 ∩ suc ∪ (𝐴 ∪ 𝐵)) = 𝐵)
174	172, 173	sylib 208	. . . 4 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐵 ∩ suc ∪ (𝐴 ∪ 𝐵)) = 𝐵)
175	174	fveq2d 6195	. . 3 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → (𝐹‘(𝐵 ∩ suc ∪ (𝐴 ∪ 𝐵))) = (𝐹‘𝐵))
176	171, 175	eqeq12d 2637	. 2 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘(𝐴 ∩ suc ∪ (𝐴 ∪ 𝐵))) = (𝐹‘(𝐵 ∩ suc ∪ (𝐴 ∪ 𝐵))) ↔ (𝐹‘𝐴) = (𝐹‘𝐵)))
177	170, 174	eqeq12d 2637	. 2 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐴 ∩ suc ∪ (𝐴 ∪ 𝐵)) = (𝐵 ∩ suc ∪ (𝐴 ∪ 𝐵)) ↔ 𝐴 = 𝐵))
178	163, 176, 177	3imtr3d 282	1 ⊢ ((𝐴 ∈ (𝒫 ω ∩ Fin) ∧ 𝐵 ∈ (𝒫 ω ∩ Fin)) → ((𝐹‘𝐴) = (𝐹‘𝐵) → 𝐴 = 𝐵))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 {csn 4177 ∪ cuni 4436 ∪ ciun 4520 ↦ cmpt 4729 × cxp 5112 Ord word 5722 Oncon0 5723 suc csuc 5725 ‘cfv 5888 (class class class)co 6650 ωcom 7065 +_𝑜 coa 7557 Fincfn 7955 cardccrd 8761

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-cda 8990

This theorem is referenced by: ackbij1lem17 9058

W3C validator