Theorem nnawordex 7717

Description: Equivalence for weak ordering of natural numbers. (Contributed by NM, 8-Nov-2002.) (Revised by Mario Carneiro, 15-Nov-2014.)

Assertion

Ref	Expression
nnawordex	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem nnawordex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 792	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ ω)
2		nnon 7071	. . . . . . . 8 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On)
3	1, 2	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ On)
4		simpll 790	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ∈ ω)
5		nnaword2 7710	. . . . . . . 8 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → 𝐵 ⊆ (𝐴 +𝑜 𝐵))
6	1, 4, 5	syl2anc 693	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ⊆ (𝐴 +𝑜 𝐵))
7		oveq2 6658	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 𝐵))
8	7	sseq2d 3633	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 𝐵)))
9	8	elrab 3363	. . . . . . 7 ⊢ (𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ↔ (𝐵 ∈ On ∧ 𝐵 ⊆ (𝐴 +𝑜 𝐵)))
10	3, 6, 9	sylanbrc 698	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
11		intss1 4492	. . . . . 6 ⊢ (𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ 𝐵)
12	10, 11	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ 𝐵)
13		ssrab2 3687	. . . . . . . 8 ⊢ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ On
14		ne0i 3921	. . . . . . . . 9 ⊢ (𝐵 ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ≠ ∅)
15	10, 14	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ≠ ∅)
16		oninton 7000	. . . . . . . 8 ⊢ (({𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ≠ ∅) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ On)
17	13, 15, 16	sylancr 695	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ On)
18		eloni 5733	. . . . . . 7 ⊢ (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ On → Ord ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
19	17, 18	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → Ord ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
20		ordom 7074	. . . . . 6 ⊢ Ord ω
21		ordtr2 5768	. . . . . 6 ⊢ ((Ord ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∧ Ord ω) → ((∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ 𝐵 ∧ 𝐵 ∈ ω) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ ω))
22	19, 20, 21	sylancl 694	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ((∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ 𝐵 ∧ 𝐵 ∈ ω) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ ω))
23	12, 1, 22	mp2and 715	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ ω)
24		nna0 7684	. . . . . . . . 9 ⊢ (𝐴 ∈ ω → (𝐴 +𝑜 ∅) = 𝐴)
25	24	ad2antrr 762	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (𝐴 +𝑜 ∅) = 𝐴)
26		simpr 477	. . . . . . . 8 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐴 ⊆ 𝐵)
27	25, 26	eqsstrd 3639	. . . . . . 7 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (𝐴 +𝑜 ∅) ⊆ 𝐵)
28		oveq2 6658	. . . . . . . 8 ⊢ (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = ∅ → (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) = (𝐴 +𝑜 ∅))
29	28	sseq1d 3632	. . . . . . 7 ⊢ (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = ∅ → ((𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) ⊆ 𝐵 ↔ (𝐴 +𝑜 ∅) ⊆ 𝐵))
30	27, 29	syl5ibrcom 237	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = ∅ → (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) ⊆ 𝐵))
31		simprr 796	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)
32	31	oveq2d 6666	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) = (𝐴 +𝑜 suc 𝑥))
33	4	adantr 481	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → 𝐴 ∈ ω)
34		simprl 794	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → 𝑥 ∈ ω)
35		nnasuc 7686	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 suc 𝑥) = suc (𝐴 +𝑜 𝑥))
36	33, 34, 35	syl2anc 693	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → (𝐴 +𝑜 suc 𝑥) = suc (𝐴 +𝑜 𝑥))
37	32, 36	eqtrd 2656	. . . . . . . 8 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) = suc (𝐴 +𝑜 𝑥))
38		nnord 7073	. . . . . . . . . . 11 ⊢ (𝐵 ∈ ω → Ord 𝐵)
39	1, 38	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → Ord 𝐵)
40	39	adantr 481	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → Ord 𝐵)
41		nnon 7071	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ ω → 𝑥 ∈ On)
42	41	adantr 481	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥) → 𝑥 ∈ On)
43		vex 3203	. . . . . . . . . . . . . 14 ⊢ 𝑥 ∈ V
44	43	sucid 5804	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ suc 𝑥
45		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)
46	44, 45	syl5eleqr 2708	. . . . . . . . . . . 12 ⊢ ((𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥) → 𝑥 ∈ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
47		oveq2 6658	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 𝑥))
48	47	sseq2d 3633	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 𝑥)))
49	48	onnminsb 7004	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ On → (𝑥 ∈ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑥)))
50	42, 46, 49	sylc 65	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥) → ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑥))
51	50	adantl 482	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑥))
52		nnacl 7691	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 𝑥) ∈ ω)
53	33, 34, 52	syl2anc 693	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → (𝐴 +𝑜 𝑥) ∈ ω)
54		nnord 7073	. . . . . . . . . . . . 13 ⊢ ((𝐴 +𝑜 𝑥) ∈ ω → Ord (𝐴 +𝑜 𝑥))
55	53, 54	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → Ord (𝐴 +𝑜 𝑥))
56		ordtri1 5756	. . . . . . . . . . . 12 ⊢ ((Ord 𝐵 ∧ Ord (𝐴 +𝑜 𝑥)) → (𝐵 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐵))
57	40, 55, 56	syl2anc 693	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → (𝐵 ⊆ (𝐴 +𝑜 𝑥) ↔ ¬ (𝐴 +𝑜 𝑥) ∈ 𝐵))
58	57	con2bid 344	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → ((𝐴 +𝑜 𝑥) ∈ 𝐵 ↔ ¬ 𝐵 ⊆ (𝐴 +𝑜 𝑥)))
59	51, 58	mpbird 247	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → (𝐴 +𝑜 𝑥) ∈ 𝐵)
60		ordsucss 7018	. . . . . . . . 9 ⊢ (Ord 𝐵 → ((𝐴 +𝑜 𝑥) ∈ 𝐵 → suc (𝐴 +𝑜 𝑥) ⊆ 𝐵))
61	40, 59, 60	sylc 65	. . . . . . . 8 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → suc (𝐴 +𝑜 𝑥) ⊆ 𝐵)
62	37, 61	eqsstrd 3639	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) ∧ (𝑥 ∈ ω ∧ ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥)) → (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) ⊆ 𝐵)
63	62	rexlimdvaa 3032	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (∃𝑥 ∈ ω ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥 → (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) ⊆ 𝐵))
64		nn0suc 7090	. . . . . . 7 ⊢ (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ ω → (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = ∅ ∨ ∃𝑥 ∈ ω ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥))
65	23, 64	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = ∅ ∨ ∃𝑥 ∈ ω ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} = suc 𝑥))
66	30, 63, 65	mpjaod 396	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) ⊆ 𝐵)
67		onint 6995	. . . . . . 7 ⊢ (({𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ⊆ On ∧ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ≠ ∅) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
68	13, 15, 67	sylancr 695	. . . . . 6 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
69		nfrab1 3122	. . . . . . . . 9 ⊢ Ⅎ𝑦{𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
70	69	nfint 4486	. . . . . . . 8 ⊢ Ⅎ𝑦∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}
71		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑦On
72		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐵
73		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐴
74		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑦 +𝑜
75	73, 74, 70	nfov 6676	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
76	72, 75	nfss 3596	. . . . . . . 8 ⊢ Ⅎ𝑦 𝐵 ⊆ (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})
77		oveq2 6658	. . . . . . . . 9 ⊢ (𝑦 = ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → (𝐴 +𝑜 𝑦) = (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}))
78	77	sseq2d 3633	. . . . . . . 8 ⊢ (𝑦 = ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → (𝐵 ⊆ (𝐴 +𝑜 𝑦) ↔ 𝐵 ⊆ (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})))
79	70, 71, 76, 78	elrabf 3360	. . . . . . 7 ⊢ (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ↔ (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ On ∧ 𝐵 ⊆ (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)})))
80	79	simprbi 480	. . . . . 6 ⊢ (∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → 𝐵 ⊆ (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}))
81	68, 80	syl 17	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → 𝐵 ⊆ (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}))
82	66, 81	eqssd 3620	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) = 𝐵)
83		oveq2 6658	. . . . . 6 ⊢ (𝑥 = ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → (𝐴 +𝑜 𝑥) = (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}))
84	83	eqeq1d 2624	. . . . 5 ⊢ (𝑥 = ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} → ((𝐴 +𝑜 𝑥) = 𝐵 ↔ (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) = 𝐵))
85	84	rspcev 3309	. . . 4 ⊢ ((∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)} ∈ ω ∧ (𝐴 +𝑜 ∩ {𝑦 ∈ On ∣ 𝐵 ⊆ (𝐴 +𝑜 𝑦)}) = 𝐵) → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵)
86	23, 82, 85	syl2anc 693	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐴 ⊆ 𝐵) → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵)
87	86	ex 450	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))
88		nnaword1 7709	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
89	88	adantlr 751	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑥 ∈ ω) → 𝐴 ⊆ (𝐴 +𝑜 𝑥))
90		sseq2 3627	. . . 4 ⊢ ((𝐴 +𝑜 𝑥) = 𝐵 → (𝐴 ⊆ (𝐴 +𝑜 𝑥) ↔ 𝐴 ⊆ 𝐵))
91	89, 90	syl5ibcom 235	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝑥 ∈ ω) → ((𝐴 +𝑜 𝑥) = 𝐵 → 𝐴 ⊆ 𝐵))
92	91	rexlimdva 3031	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵 → 𝐴 ⊆ 𝐵))
93	87, 92	impbid 202	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ↔ ∃𝑥 ∈ ω (𝐴 +𝑜 𝑥) = 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913  {crab 2916   ⊆ wss 3574  ∅c0 3915  ∩ cint 4475  Ord word 5722  Oncon0 5723  suc csuc 5725  (class class class)co 6650  ωcom 7065   +_𝑜 coa 7557

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-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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-oadd 7564

This theorem is referenced by:  nnaordex  7718  unfilem1  8224  hashdom  13168