Theorem pwcfsdom 9405

Description: A corollary of Konig's Theorem konigth 9391. Theorem 11.28 of [TakeutiZaring] p. 108. (Contributed by Mario Carneiro, 20-Mar-2013.)

Hypothesis

Ref	Expression
pwcfsdom.1	⊢ 𝐻 = (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑦)))

Assertion

Ref	Expression
pwcfsdom	⊢ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))

Distinct variable group:   𝐴,𝑓,𝑦

Allowed substitution hints:   𝐻(𝑦,𝑓)

Proof of Theorem pwcfsdom

Dummy variables 𝑤 𝑧 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onzsl 7046	. . . 4 ⊢ (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
2	1	biimpi 206	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
3		cfom 9086	. . . . . . 7 ⊢ (cf‘ω) = ω
4		aleph0 8889	. . . . . . . 8 ⊢ (ℵ‘∅) = ω
5	4	fveq2i 6194	. . . . . . 7 ⊢ (cf‘(ℵ‘∅)) = (cf‘ω)
6	3, 5, 4	3eqtr4i 2654	. . . . . 6 ⊢ (cf‘(ℵ‘∅)) = (ℵ‘∅)
7		fveq2 6191	. . . . . . 7 ⊢ (𝐴 = ∅ → (ℵ‘𝐴) = (ℵ‘∅))
8	7	fveq2d 6195	. . . . . 6 ⊢ (𝐴 = ∅ → (cf‘(ℵ‘𝐴)) = (cf‘(ℵ‘∅)))
9	6, 8, 7	3eqtr4a 2682	. . . . 5 ⊢ (𝐴 = ∅ → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
10		fvex 6201	. . . . . . . . 9 ⊢ (ℵ‘𝐴) ∈ V
11	10	canth2 8113	. . . . . . . 8 ⊢ (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
12	10	pw2en 8067	. . . . . . . 8 ⊢ 𝒫 (ℵ‘𝐴) ≈ (2𝑜 ↑𝑚 (ℵ‘𝐴))
13		sdomentr 8094	. . . . . . . 8 ⊢ (((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≈ (2𝑜 ↑𝑚 (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (2𝑜 ↑𝑚 (ℵ‘𝐴)))
14	11, 12, 13	mp2an 708	. . . . . . 7 ⊢ (ℵ‘𝐴) ≺ (2𝑜 ↑𝑚 (ℵ‘𝐴))
15		alephon 8892	. . . . . . . . 9 ⊢ (ℵ‘𝐴) ∈ On
16		alephgeom 8905	. . . . . . . . . 10 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
17		omelon 8543	. . . . . . . . . . . 12 ⊢ ω ∈ On
18		2onn 7720	. . . . . . . . . . . 12 ⊢ 2𝑜 ∈ ω
19		onelss 5766	. . . . . . . . . . . 12 ⊢ (ω ∈ On → (2𝑜 ∈ ω → 2𝑜 ⊆ ω))
20	17, 18, 19	mp2 9	. . . . . . . . . . 11 ⊢ 2𝑜 ⊆ ω
21		sstr 3611	. . . . . . . . . . 11 ⊢ ((2𝑜 ⊆ ω ∧ ω ⊆ (ℵ‘𝐴)) → 2𝑜 ⊆ (ℵ‘𝐴))
22	20, 21	mpan 706	. . . . . . . . . 10 ⊢ (ω ⊆ (ℵ‘𝐴) → 2𝑜 ⊆ (ℵ‘𝐴))
23	16, 22	sylbi 207	. . . . . . . . 9 ⊢ (𝐴 ∈ On → 2𝑜 ⊆ (ℵ‘𝐴))
24		ssdomg 8001	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) ∈ On → (2𝑜 ⊆ (ℵ‘𝐴) → 2𝑜 ≼ (ℵ‘𝐴)))
25	15, 23, 24	mpsyl 68	. . . . . . . 8 ⊢ (𝐴 ∈ On → 2𝑜 ≼ (ℵ‘𝐴))
26		mapdom1 8125	. . . . . . . 8 ⊢ (2𝑜 ≼ (ℵ‘𝐴) → (2𝑜 ↑𝑚 (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)))
27	25, 26	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ On → (2𝑜 ↑𝑚 (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)))
28		sdomdomtr 8093	. . . . . . 7 ⊢ (((ℵ‘𝐴) ≺ (2𝑜 ↑𝑚 (ℵ‘𝐴)) ∧ (2𝑜 ↑𝑚 (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)))
29	14, 27, 28	sylancr 695	. . . . . 6 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)))
30		oveq2 6658	. . . . . . 7 ⊢ ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) = ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴)))
31	30	breq2d 4665	. . . . . 6 ⊢ ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → ((ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) ↔ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (ℵ‘𝐴))))
32	29, 31	syl5ibrcom 237	. . . . 5 ⊢ (𝐴 ∈ On → ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
33	9, 32	syl5 34	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 = ∅ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
34		alephreg 9404	. . . . . . 7 ⊢ (cf‘(ℵ‘suc 𝑥)) = (ℵ‘suc 𝑥)
35		fveq2 6191	. . . . . . . 8 ⊢ (𝐴 = suc 𝑥 → (ℵ‘𝐴) = (ℵ‘suc 𝑥))
36	35	fveq2d 6195	. . . . . . 7 ⊢ (𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (cf‘(ℵ‘suc 𝑥)))
37	34, 36, 35	3eqtr4a 2682	. . . . . 6 ⊢ (𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
38	37	rexlimivw 3029	. . . . 5 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
39	38, 32	syl5 34	. . . 4 ⊢ (𝐴 ∈ On → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
40		cfsmo 9093	. . . . . 6 ⊢ ((ℵ‘𝐴) ∈ On → ∃𝑓(𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤)))
41		limelon 5788	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
42		ffn 6045	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → 𝑓 Fn (cf‘(ℵ‘𝐴)))
43		fnrnfv 6242	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 Fn (cf‘(ℵ‘𝐴)) → ran 𝑓 = {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓‘𝑥)})
44	43	unieqd 4446	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fn (cf‘(ℵ‘𝐴)) → ∪ ran 𝑓 = ∪ {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓‘𝑥)})
45	42, 44	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∪ ran 𝑓 = ∪ {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓‘𝑥)})
46		fvex 6201	. . . . . . . . . . . . . . . 16 ⊢ (𝑓‘𝑥) ∈ V
47	46	dfiun2 4554	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) = ∪ {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓‘𝑥)}
48	45, 47	syl6eqr 2674	. . . . . . . . . . . . . 14 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∪ ran 𝑓 = ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥))
49	48	ad2antrl 764	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → ∪ ran 𝑓 = ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥))
50		fnfvelrn 6356	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓 Fn (cf‘(ℵ‘𝐴)) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑤) ∈ ran 𝑓)
51	42, 50	sylan 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑤) ∈ ran 𝑓)
52		sseq2 3627	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = (𝑓‘𝑤) → (𝑧 ⊆ 𝑦 ↔ 𝑧 ⊆ (𝑓‘𝑤)))
53	52	rspcev 3309	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓‘𝑤) ∈ ran 𝑓 ∧ 𝑧 ⊆ (𝑓‘𝑤)) → ∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦)
54	51, 53	sylan 488	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) ∧ 𝑧 ⊆ (𝑓‘𝑤)) → ∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦)
55	54	ex 450	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑧 ⊆ (𝑓‘𝑤) → ∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦))
56	55	rexlimdva 3031	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤) → ∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦))
57	56	ralimdv 2963	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤) → ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦))
58	57	imp 445	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤)) → ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦)
59	58	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦)
60		alephislim 8906	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On ↔ Lim (ℵ‘𝐴))
61	60	biimpi 206	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ On → Lim (ℵ‘𝐴))
62		frn 6053	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ran 𝑓 ⊆ (ℵ‘𝐴))
63	62	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤)) → ran 𝑓 ⊆ (ℵ‘𝐴))
64		coflim 9083	. . . . . . . . . . . . . . 15 ⊢ ((Lim (ℵ‘𝐴) ∧ ran 𝑓 ⊆ (ℵ‘𝐴)) → (∪ ran 𝑓 = (ℵ‘𝐴) ↔ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦))
65	61, 63, 64	syl2an 494	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → (∪ ran 𝑓 = (ℵ‘𝐴) ↔ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦))
66	59, 65	mpbird 247	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → ∪ ran 𝑓 = (ℵ‘𝐴))
67	49, 66	eqtr3d 2658	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) = (ℵ‘𝐴))
68		ffvelrn 6357	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑥) ∈ (ℵ‘𝐴))
69	15	oneli 5835	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑥) ∈ (ℵ‘𝐴) → (𝑓‘𝑥) ∈ On)
70		harcard 8804	. . . . . . . . . . . . . . . . . . 19 ⊢ (card‘(har‘(𝑓‘𝑥))) = (har‘(𝑓‘𝑥))
71		iscard 8801	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((card‘(har‘(𝑓‘𝑥))) = (har‘(𝑓‘𝑥)) ↔ ((har‘(𝑓‘𝑥)) ∈ On ∧ ∀𝑦 ∈ (har‘(𝑓‘𝑥))𝑦 ≺ (har‘(𝑓‘𝑥))))
72	71	simprbi 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((card‘(har‘(𝑓‘𝑥))) = (har‘(𝑓‘𝑥)) → ∀𝑦 ∈ (har‘(𝑓‘𝑥))𝑦 ≺ (har‘(𝑓‘𝑥)))
73	70, 72	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ∀𝑦 ∈ (har‘(𝑓‘𝑥))𝑦 ≺ (har‘(𝑓‘𝑥))
74		domrefg 7990	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝑥) ∈ V → (𝑓‘𝑥) ≼ (𝑓‘𝑥))
75	46, 74	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓‘𝑥) ≼ (𝑓‘𝑥)
76		elharval 8468	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝑥) ∈ (har‘(𝑓‘𝑥)) ↔ ((𝑓‘𝑥) ∈ On ∧ (𝑓‘𝑥) ≼ (𝑓‘𝑥)))
77	76	biimpri 218	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓‘𝑥) ∈ On ∧ (𝑓‘𝑥) ≼ (𝑓‘𝑥)) → (𝑓‘𝑥) ∈ (har‘(𝑓‘𝑥)))
78	75, 77	mpan2 707	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑥) ∈ On → (𝑓‘𝑥) ∈ (har‘(𝑓‘𝑥)))
79		breq1 4656	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑓‘𝑥) → (𝑦 ≺ (har‘(𝑓‘𝑥)) ↔ (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
80	79	rspccv 3306	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑦 ∈ (har‘(𝑓‘𝑥))𝑦 ≺ (har‘(𝑓‘𝑥)) → ((𝑓‘𝑥) ∈ (har‘(𝑓‘𝑥)) → (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
81	73, 78, 80	mpsyl 68	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑥) ∈ On → (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥)))
82	68, 69, 81	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥)))
83		harcl 8466	. . . . . . . . . . . . . . . . . . 19 ⊢ (har‘(𝑓‘𝑥)) ∈ On
84		fveq2 6191	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 = 𝑥 → (𝑓‘𝑦) = (𝑓‘𝑥))
85	84	fveq2d 6195	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → (har‘(𝑓‘𝑦)) = (har‘(𝑓‘𝑥)))
86		pwcfsdom.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐻 = (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑦)))
87	85, 86	fvmptg 6280	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (cf‘(ℵ‘𝐴)) ∧ (har‘(𝑓‘𝑥)) ∈ On) → (𝐻‘𝑥) = (har‘(𝑓‘𝑥)))
88	83, 87	mpan2 707	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (𝐻‘𝑥) = (har‘(𝑓‘𝑥)))
89	88	breq2d 4665	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (cf‘(ℵ‘𝐴)) → ((𝑓‘𝑥) ≺ (𝐻‘𝑥) ↔ (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
90	89	adantl 482	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → ((𝑓‘𝑥) ≺ (𝐻‘𝑥) ↔ (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
91	82, 90	mpbird 247	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑥) ≺ (𝐻‘𝑥))
92	91	ralrimiva 2966	. . . . . . . . . . . . . 14 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ (𝐻‘𝑥))
93		fvex 6201	. . . . . . . . . . . . . . 15 ⊢ (cf‘(ℵ‘𝐴)) ∈ V
94		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) = ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥)
95		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) = X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥)
96	93, 94, 95	konigth 9391	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ (𝐻‘𝑥) → ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥))
97	92, 96	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥))
98	97	ad2antrl 764	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥))
99	67, 98	eqbrtrrd 4677	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → (ℵ‘𝐴) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥))
100	41, 99	sylan 488	. . . . . . . . . 10 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → (ℵ‘𝐴) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥))
101		ovex 6678	. . . . . . . . . . . 12 ⊢ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) ∈ V
102	68	ex 450	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (𝑓‘𝑥) ∈ (ℵ‘𝐴)))
103		alephlim 8890	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) = ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦))
104	103	eleq2d 2687	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓‘𝑥) ∈ (ℵ‘𝐴) ↔ (𝑓‘𝑥) ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦)))
105		eliun 4524	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑥) ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) ↔ ∃𝑦 ∈ 𝐴 (𝑓‘𝑥) ∈ (ℵ‘𝑦))
106		alephcard 8893	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)
107	106	eleq2i 2693	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓‘𝑥) ∈ (card‘(ℵ‘𝑦)) ↔ (𝑓‘𝑥) ∈ (ℵ‘𝑦))
108		cardsdomelir 8799	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓‘𝑥) ∈ (card‘(ℵ‘𝑦)) → (𝑓‘𝑥) ≺ (ℵ‘𝑦))
109	107, 108	sylbir 225	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓‘𝑥) ∈ (ℵ‘𝑦) → (𝑓‘𝑥) ≺ (ℵ‘𝑦))
110		elharval 8468	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)) ↔ ((ℵ‘𝑦) ∈ On ∧ (ℵ‘𝑦) ≼ (𝑓‘𝑥)))
111	110	simprbi 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)) → (ℵ‘𝑦) ≼ (𝑓‘𝑥))
112		domnsym 8086	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((ℵ‘𝑦) ≼ (𝑓‘𝑥) → ¬ (𝑓‘𝑥) ≺ (ℵ‘𝑦))
113	111, 112	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)) → ¬ (𝑓‘𝑥) ≺ (ℵ‘𝑦))
114	113	con2i 134	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑓‘𝑥) ≺ (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)))
115		alephon 8892	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (ℵ‘𝑦) ∈ On
116		ontri1 5757	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((har‘(𝑓‘𝑥)) ∈ On ∧ (ℵ‘𝑦) ∈ On) → ((har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦) ↔ ¬ (ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥))))
117	83, 115, 116	mp2an 708	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦) ↔ ¬ (ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)))
118	114, 117	sylibr 224	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓‘𝑥) ≺ (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦))
119	109, 118	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓‘𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦))
120		alephord2i 8900	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
121	120	imp 445	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → (ℵ‘𝑦) ∈ (ℵ‘𝐴))
122		ontr2 5772	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((har‘(𝑓‘𝑥)) ∈ On ∧ (ℵ‘𝐴) ∈ On) → (((har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝐴)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
123	83, 15, 122	mp2an 708	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝐴)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴))
124	119, 121, 123	syl2anr 495	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑓‘𝑥) ∈ (ℵ‘𝑦)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴))
125	124	exp31 630	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → ((𝑓‘𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴))))
126	125	rexlimdv 3030	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ On → (∃𝑦 ∈ 𝐴 (𝑓‘𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
127	105, 126	syl5bi 232	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ On → ((𝑓‘𝑥) ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
128	41, 127	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓‘𝑥) ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
129	104, 128	sylbid 230	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓‘𝑥) ∈ (ℵ‘𝐴) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
130	102, 129	sylan9r 690	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
131	130	imp 445	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴))
132	85	cbvmptv 4750	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑦))) = (𝑥 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑥)))
133	86, 132	eqtri 2644	. . . . . . . . . . . . . 14 ⊢ 𝐻 = (𝑥 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑥)))
134	131, 133	fmptd 6385	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → 𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴))
135		ffvelrn 6357	. . . . . . . . . . . . . . 15 ⊢ ((𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝐻‘𝑥) ∈ (ℵ‘𝐴))
136		onelss 5766	. . . . . . . . . . . . . . 15 ⊢ ((ℵ‘𝐴) ∈ On → ((𝐻‘𝑥) ∈ (ℵ‘𝐴) → (𝐻‘𝑥) ⊆ (ℵ‘𝐴)))
137	15, 135, 136	mpsyl 68	. . . . . . . . . . . . . 14 ⊢ ((𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝐻‘𝑥) ⊆ (ℵ‘𝐴))
138	137	ralrimiva 2966	. . . . . . . . . . . . 13 ⊢ (𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ (ℵ‘𝐴))
139		ss2ixp 7921	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ (ℵ‘𝐴) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ X𝑥 ∈ (cf‘(ℵ‘𝐴))(ℵ‘𝐴))
140	93, 10	ixpconst 7918	. . . . . . . . . . . . . 14 ⊢ X𝑥 ∈ (cf‘(ℵ‘𝐴))(ℵ‘𝐴) = ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))
141	139, 140	syl6sseq 3651	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ (ℵ‘𝐴) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
142	134, 138, 141	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
143		ssdomg 8001	. . . . . . . . . . . 12 ⊢ (((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) ∈ V → (X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ≼ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
144	101, 142, 143	mpsyl 68	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ≼ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
145	144	adantrr 753	. . . . . . . . . 10 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ≼ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
146		sdomdomtr 8093	. . . . . . . . . 10 ⊢ (((ℵ‘𝐴) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ∧ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ≼ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
147	100, 145, 146	syl2anc 693	. . . . . . . . 9 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
148	147	expcom 451	. . . . . . . 8 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤)) → ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
149	148	3adant2 1080	. . . . . . 7 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤)) → ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
150	149	exlimiv 1858	. . . . . 6 ⊢ (∃𝑓(𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤)) → ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
151	15, 40, 150	mp2b 10	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
152	151	a1i 11	. . . 4 ⊢ (𝐴 ∈ On → ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
153	33, 39, 152	3jaod 1392	. . 3 ⊢ (𝐴 ∈ On → ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))))
154	2, 153	mpd 15	. 2 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
155		alephfnon 8888	. . . . 5 ⊢ ℵ Fn On
156		fndm 5990	. . . . 5 ⊢ (ℵ Fn On → dom ℵ = On)
157	155, 156	ax-mp 5	. . . 4 ⊢ dom ℵ = On
158	157	eleq2i 2693	. . 3 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
159		ndmfv 6218	. . . 4 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
160		1n0 7575	. . . . . 6 ⊢ 1𝑜 ≠ ∅
161		1on 7567	. . . . . . . 8 ⊢ 1𝑜 ∈ On
162	161	elexi 3213	. . . . . . 7 ⊢ 1𝑜 ∈ V
163	162	0sdom 8091	. . . . . 6 ⊢ (∅ ≺ 1𝑜 ↔ 1𝑜 ≠ ∅)
164	160, 163	mpbir 221	. . . . 5 ⊢ ∅ ≺ 1𝑜
165		id 22	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
166		fveq2 6191	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(ℵ‘𝐴)) = (cf‘∅))
167		cf0 9073	. . . . . . . . 9 ⊢ (cf‘∅) = ∅
168	166, 167	syl6eq 2672	. . . . . . . 8 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(ℵ‘𝐴)) = ∅)
169	165, 168	oveq12d 6668	. . . . . . 7 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) = (∅ ↑𝑚 ∅))
170		0ex 4790	. . . . . . . 8 ⊢ ∅ ∈ V
171		map0e 7895	. . . . . . . 8 ⊢ (∅ ∈ V → (∅ ↑𝑚 ∅) = 1𝑜)
172	170, 171	ax-mp 5	. . . . . . 7 ⊢ (∅ ↑𝑚 ∅) = 1𝑜
173	169, 172	syl6eq 2672	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) = 1𝑜)
174	165, 173	breq12d 4666	. . . . 5 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))) ↔ ∅ ≺ 1𝑜))
175	164, 174	mpbiri 248	. . . 4 ⊢ ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
176	159, 175	syl 17	. . 3 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
177	158, 176	sylnbir 321	. 2 ⊢ (¬ 𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴))))
178	154, 177	pm2.61i 176	1 ⊢ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑𝑚 (cf‘(ℵ‘𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∨ w3o 1036   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158  ∪ cuni 4436  ∪ ciun 4520   class class class wbr 4653   ↦ cmpt 4729  dom cdm 5114  ran crn 5115  Oncon0 5723  Lim wlim 5724  suc csuc 5725   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ωcom 7065  Smo wsmo 7442  1_𝑜c1o 7553  2_𝑜c2o 7554   ↑_𝑚 cmap 7857  Xcixp 7908   ≈ cen 7952   ≼ cdom 7953   ≺ csdm 7954  harchar 8461  cardccrd 8761  ℵcale 8762  cfccf 8763

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  ax-inf2 8538  ax-ac2 9285

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-iin 4523  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-smo 7443  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-oi 8415  df-har 8463  df-card 8765  df-aleph 8766  df-cf 8767  df-acn 8768  df-ac 8939

This theorem is referenced by:  cfpwsdom  9406  tskcard  9603  bj-pwcfsdom  33022