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Theorem pwcfsdom 10270

Description: A corollary of Konig's Theorem konigth 10256. 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 7668	. . . 4 ⊢ (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
2	1	biimpi 215	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
3		cfom 9951	. . . . . . 7 ⊢ (cf‘ω) = ω
4		aleph0 9753	. . . . . . . 8 ⊢ (ℵ‘∅) = ω
5	4	fveq2i 6759	. . . . . . 7 ⊢ (cf‘(ℵ‘∅)) = (cf‘ω)
6	3, 5, 4	3eqtr4i 2776	. . . . . 6 ⊢ (cf‘(ℵ‘∅)) = (ℵ‘∅)
7		2fveq3 6761	. . . . . 6 ⊢ (𝐴 = ∅ → (cf‘(ℵ‘𝐴)) = (cf‘(ℵ‘∅)))
8		fveq2 6756	. . . . . 6 ⊢ (𝐴 = ∅ → (ℵ‘𝐴) = (ℵ‘∅))
9	6, 7, 8	3eqtr4a 2805	. . . . 5 ⊢ (𝐴 = ∅ → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
10		fvex 6769	. . . . . . . . 9 ⊢ (ℵ‘𝐴) ∈ V
11	10	canth2 8866	. . . . . . . 8 ⊢ (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
12	10	pw2en 8819	. . . . . . . 8 ⊢ 𝒫 (ℵ‘𝐴) ≈ (2_o ↑_m (ℵ‘𝐴))
13		sdomentr 8847	. . . . . . . 8 ⊢ (((ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴) ∧ 𝒫 (ℵ‘𝐴) ≈ (2_o ↑_m (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ (2_o ↑_m (ℵ‘𝐴)))
14	11, 12, 13	mp2an 688	. . . . . . 7 ⊢ (ℵ‘𝐴) ≺ (2_o ↑_m (ℵ‘𝐴))
15		alephon 9756	. . . . . . . . 9 ⊢ (ℵ‘𝐴) ∈ On
16		alephgeom 9769	. . . . . . . . . 10 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
17		omelon 9334	. . . . . . . . . . . 12 ⊢ ω ∈ On
18		2onn 8433	. . . . . . . . . . . 12 ⊢ 2_o ∈ ω
19		onelss 6293	. . . . . . . . . . . 12 ⊢ (ω ∈ On → (2_o ∈ ω → 2_o ⊆ ω))
20	17, 18, 19	mp2 9	. . . . . . . . . . 11 ⊢ 2_o ⊆ ω
21		sstr 3925	. . . . . . . . . . 11 ⊢ ((2_o ⊆ ω ∧ ω ⊆ (ℵ‘𝐴)) → 2_o ⊆ (ℵ‘𝐴))
22	20, 21	mpan 686	. . . . . . . . . 10 ⊢ (ω ⊆ (ℵ‘𝐴) → 2_o ⊆ (ℵ‘𝐴))
23	16, 22	sylbi 216	. . . . . . . . 9 ⊢ (𝐴 ∈ On → 2_o ⊆ (ℵ‘𝐴))
24		ssdomg 8741	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) ∈ On → (2_o ⊆ (ℵ‘𝐴) → 2_o ≼ (ℵ‘𝐴)))
25	15, 23, 24	mpsyl 68	. . . . . . . 8 ⊢ (𝐴 ∈ On → 2_o ≼ (ℵ‘𝐴))
26		mapdom1 8878	. . . . . . . 8 ⊢ (2_o ≼ (ℵ‘𝐴) → (2_o ↑_m (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑_m (ℵ‘𝐴)))
27	25, 26	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ On → (2_o ↑_m (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑_m (ℵ‘𝐴)))
28		sdomdomtr 8846	. . . . . . 7 ⊢ (((ℵ‘𝐴) ≺ (2_o ↑_m (ℵ‘𝐴)) ∧ (2_o ↑_m (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑_m (ℵ‘𝐴))) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (ℵ‘𝐴)))
29	14, 27, 28	sylancr 586	. . . . . 6 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (ℵ‘𝐴)))
30		oveq2 7263	. . . . . . 7 ⊢ ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) = ((ℵ‘𝐴) ↑_m (ℵ‘𝐴)))
31	30	breq2d 5082	. . . . . 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 246	. . . . 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 10269	. . . . . . 7 ⊢ (cf‘(ℵ‘suc 𝑥)) = (ℵ‘suc 𝑥)
35		2fveq3 6761	. . . . . . 7 ⊢ (𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (cf‘(ℵ‘suc 𝑥)))
36		fveq2 6756	. . . . . . 7 ⊢ (𝐴 = suc 𝑥 → (ℵ‘𝐴) = (ℵ‘suc 𝑥))
37	34, 35, 36	3eqtr4a 2805	. . . . . 6 ⊢ (𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
38	37	rexlimivw 3210	. . . . 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 9958	. . . . . 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 6314	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
42		ffn 6584	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → 𝑓 Fn (cf‘(ℵ‘𝐴)))
43		fnrnfv 6811	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 Fn (cf‘(ℵ‘𝐴)) → ran 𝑓 = {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓‘𝑥)})
44	43	unieqd 4850	. . . . . . . . . . . . . . . 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 6769	. . . . . . . . . . . . . . . 16 ⊢ (𝑓‘𝑥) ∈ V
47	46	dfiun2 4959	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) = ∪ {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓‘𝑥)}
48	45, 47	eqtr4di 2797	. . . . . . . . . . . . . 14 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∪ ran 𝑓 = ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥))
49	48	ad2antrl 724	. . . . . . . . . . . . 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 6940	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 Fn (cf‘(ℵ‘𝐴)) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑤) ∈ ran 𝑓)
51	42, 50	sylan 579	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑤) ∈ ran 𝑓)
52		sseq2 3943	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑓‘𝑤) → (𝑧 ⊆ 𝑦 ↔ 𝑧 ⊆ (𝑓‘𝑤)))
53	52	rspcev 3552	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓‘𝑤) ∈ ran 𝑓 ∧ 𝑧 ⊆ (𝑓‘𝑤)) → ∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦)
54	51, 53	sylan 579	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) ∧ 𝑧 ⊆ (𝑓‘𝑤)) → ∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦)
55	54	rexlimdva2 3215	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤) → ∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦))
56	55	ralimdv 3103	. . . . . . . . . . . . . . . 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))
57	56	imp 406	. . . . . . . . . . . . . . 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)
58	57	adantl 481	. . . . . . . . . . . . . 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)
59		alephislim 9770	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On ↔ Lim (ℵ‘𝐴))
60	59	biimpi 215	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ On → Lim (ℵ‘𝐴))
61		frn 6591	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ran 𝑓 ⊆ (ℵ‘𝐴))
62	61	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤)) → ran 𝑓 ⊆ (ℵ‘𝐴))
63		coflim 9948	. . . . . . . . . . . . . . 15 ⊢ ((Lim (ℵ‘𝐴) ∧ ran 𝑓 ⊆ (ℵ‘𝐴)) → (∪ ran 𝑓 = (ℵ‘𝐴) ↔ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦))
64	60, 62, 63	syl2an 595	. . . . . . . . . . . . . 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))
65	58, 64	mpbird 256	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → ∪ ran 𝑓 = (ℵ‘𝐴))
66	49, 65	eqtr3d 2780	. . . . . . . . . . . 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))
67		ffvelrn 6941	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑥) ∈ (ℵ‘𝐴))
68	15	oneli 6359	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑥) ∈ (ℵ‘𝐴) → (𝑓‘𝑥) ∈ On)
69		harcard 9667	. . . . . . . . . . . . . . . . . . 19 ⊢ (card‘(har‘(𝑓‘𝑥))) = (har‘(𝑓‘𝑥))
70		iscard 9664	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((card‘(har‘(𝑓‘𝑥))) = (har‘(𝑓‘𝑥)) ↔ ((har‘(𝑓‘𝑥)) ∈ On ∧ ∀𝑦 ∈ (har‘(𝑓‘𝑥))𝑦 ≺ (har‘(𝑓‘𝑥))))
71	70	simprbi 496	. . . . . . . . . . . . . . . . . . 19 ⊢ ((card‘(har‘(𝑓‘𝑥))) = (har‘(𝑓‘𝑥)) → ∀𝑦 ∈ (har‘(𝑓‘𝑥))𝑦 ≺ (har‘(𝑓‘𝑥)))
72	69, 71	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ∀𝑦 ∈ (har‘(𝑓‘𝑥))𝑦 ≺ (har‘(𝑓‘𝑥))
73		domrefg 8730	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝑥) ∈ V → (𝑓‘𝑥) ≼ (𝑓‘𝑥))
74	46, 73	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓‘𝑥) ≼ (𝑓‘𝑥)
75		elharval 9250	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝑥) ∈ (har‘(𝑓‘𝑥)) ↔ ((𝑓‘𝑥) ∈ On ∧ (𝑓‘𝑥) ≼ (𝑓‘𝑥)))
76	75	biimpri 227	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓‘𝑥) ∈ On ∧ (𝑓‘𝑥) ≼ (𝑓‘𝑥)) → (𝑓‘𝑥) ∈ (har‘(𝑓‘𝑥)))
77	74, 76	mpan2 687	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑥) ∈ On → (𝑓‘𝑥) ∈ (har‘(𝑓‘𝑥)))
78		breq1 5073	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑓‘𝑥) → (𝑦 ≺ (har‘(𝑓‘𝑥)) ↔ (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
79	78	rspccv 3549	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑦 ∈ (har‘(𝑓‘𝑥))𝑦 ≺ (har‘(𝑓‘𝑥)) → ((𝑓‘𝑥) ∈ (har‘(𝑓‘𝑥)) → (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
80	72, 77, 79	mpsyl 68	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑥) ∈ On → (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥)))
81	67, 68, 80	3syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥)))
82		harcl 9248	. . . . . . . . . . . . . . . . . . 19 ⊢ (har‘(𝑓‘𝑥)) ∈ On
83		2fveq3 6761	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → (har‘(𝑓‘𝑦)) = (har‘(𝑓‘𝑥)))
84		pwcfsdom.1	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐻 = (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑦)))
85	83, 84	fvmptg 6855	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (cf‘(ℵ‘𝐴)) ∧ (har‘(𝑓‘𝑥)) ∈ On) → (𝐻‘𝑥) = (har‘(𝑓‘𝑥)))
86	82, 85	mpan2 687	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (𝐻‘𝑥) = (har‘(𝑓‘𝑥)))
87	86	breq2d 5082	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (cf‘(ℵ‘𝐴)) → ((𝑓‘𝑥) ≺ (𝐻‘𝑥) ↔ (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
88	87	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → ((𝑓‘𝑥) ≺ (𝐻‘𝑥) ↔ (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
89	81, 88	mpbird 256	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑥) ≺ (𝐻‘𝑥))
90	89	ralrimiva 3107	. . . . . . . . . . . . . 14 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ (𝐻‘𝑥))
91		fvex 6769	. . . . . . . . . . . . . . 15 ⊢ (cf‘(ℵ‘𝐴)) ∈ V
92		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) = ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥)
93		eqid 2738	. . . . . . . . . . . . . . 15 ⊢ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) = X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥)
94	91, 92, 93	konigth 10256	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ (𝐻‘𝑥) → ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥))
95	90, 94	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥))
96	95	ad2antrl 724	. . . . . . . . . . . 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))
97	66, 96	eqbrtrrd 5094	. . . . . . . . . . 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))
98	41, 97	sylan 579	. . . . . . . . . 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))
99		ovex 7288	. . . . . . . . . . . 12 ⊢ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) ∈ V
100	67	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (𝑓‘𝑥) ∈ (ℵ‘𝐴)))
101		alephlim 9754	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) = ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦))
102	101	eleq2d 2824	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓‘𝑥) ∈ (ℵ‘𝐴) ↔ (𝑓‘𝑥) ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦)))
103		eliun 4925	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑥) ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) ↔ ∃𝑦 ∈ 𝐴 (𝑓‘𝑥) ∈ (ℵ‘𝑦))
104		alephcard 9757	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)
105	104	eleq2i 2830	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓‘𝑥) ∈ (card‘(ℵ‘𝑦)) ↔ (𝑓‘𝑥) ∈ (ℵ‘𝑦))
106		cardsdomelir 9662	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓‘𝑥) ∈ (card‘(ℵ‘𝑦)) → (𝑓‘𝑥) ≺ (ℵ‘𝑦))
107	105, 106	sylbir 234	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓‘𝑥) ∈ (ℵ‘𝑦) → (𝑓‘𝑥) ≺ (ℵ‘𝑦))
108		elharval 9250	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)) ↔ ((ℵ‘𝑦) ∈ On ∧ (ℵ‘𝑦) ≼ (𝑓‘𝑥)))
109	108	simprbi 496	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)) → (ℵ‘𝑦) ≼ (𝑓‘𝑥))
110		domnsym 8839	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ℵ‘𝑦) ≼ (𝑓‘𝑥) → ¬ (𝑓‘𝑥) ≺ (ℵ‘𝑦))
111	109, 110	syl 17	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)) → ¬ (𝑓‘𝑥) ≺ (ℵ‘𝑦))
112	111	con2i 139	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓‘𝑥) ≺ (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)))
113		alephon 9756	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℵ‘𝑦) ∈ On
114		ontri1 6285	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((har‘(𝑓‘𝑥)) ∈ On ∧ (ℵ‘𝑦) ∈ On) → ((har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦) ↔ ¬ (ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥))))
115	82, 113, 114	mp2an 688	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦) ↔ ¬ (ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)))
116	112, 115	sylibr 233	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓‘𝑥) ≺ (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦))
117	107, 116	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓‘𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦))
118		alephord2i 9764	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
119	118	imp 406	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → (ℵ‘𝑦) ∈ (ℵ‘𝐴))
120		ontr2 6298	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((har‘(𝑓‘𝑥)) ∈ On ∧ (ℵ‘𝐴) ∈ On) → (((har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝐴)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
121	82, 15, 120	mp2an 688	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝐴)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴))
122	117, 119, 121	syl2anr 596	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑓‘𝑥) ∈ (ℵ‘𝑦)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴))
123	122	rexlimdva2 3215	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ On → (∃𝑦 ∈ 𝐴 (𝑓‘𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
124	103, 123	syl5bi 241	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ On → ((𝑓‘𝑥) ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
125	41, 124	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓‘𝑥) ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
126	102, 125	sylbid 239	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓‘𝑥) ∈ (ℵ‘𝐴) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
127	100, 126	sylan9r 508	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
128	127	imp 406	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴))
129	83	cbvmptv 5183	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑦))) = (𝑥 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑥)))
130	84, 129	eqtri 2766	. . . . . . . . . . . . . 14 ⊢ 𝐻 = (𝑥 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑥)))
131	128, 130	fmptd 6970	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → 𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴))
132		ffvelrn 6941	. . . . . . . . . . . . . . 15 ⊢ ((𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝐻‘𝑥) ∈ (ℵ‘𝐴))
133		onelss 6293	. . . . . . . . . . . . . . 15 ⊢ ((ℵ‘𝐴) ∈ On → ((𝐻‘𝑥) ∈ (ℵ‘𝐴) → (𝐻‘𝑥) ⊆ (ℵ‘𝐴)))
134	15, 132, 133	mpsyl 68	. . . . . . . . . . . . . 14 ⊢ ((𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝐻‘𝑥) ⊆ (ℵ‘𝐴))
135	134	ralrimiva 3107	. . . . . . . . . . . . 13 ⊢ (𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ (ℵ‘𝐴))
136		ss2ixp 8656	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ (ℵ‘𝐴) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ X𝑥 ∈ (cf‘(ℵ‘𝐴))(ℵ‘𝐴))
137	91, 10	ixpconst 8653	. . . . . . . . . . . . . 14 ⊢ X𝑥 ∈ (cf‘(ℵ‘𝐴))(ℵ‘𝐴) = ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))
138	136, 137	sseqtrdi 3967	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ (ℵ‘𝐴) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
139	131, 135, 138	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))))
140		ssdomg 8741	. . . . . . . . . . . 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)))))
141	99, 139, 140	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))))
142	141	adantrr 713	. . . . . . . . . 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))))
143		sdomdomtr 8846	. . . . . . . . . 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))))
144	98, 142, 143	syl2anc 583	. . . . . . . . 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))))
145	144	expcom 413	. . . . . . . 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)))))
146	145	3adant2 1129	. . . . . . 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)))))
147	146	exlimiv 1934	. . . . . 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)))))
148	15, 40, 147	mp2b 10	. . . . 5 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
149	148	a1i 11	. . . 4 ⊢ (𝐴 ∈ On → ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))))
150	33, 39, 149	3jaod 1426	. . 3 ⊢ (𝐴 ∈ On → ((𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))))
151	2, 150	mpd 15	. 2 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
152		alephfnon 9752	. . . . 5 ⊢ ℵ Fn On
153	152	fndmi 6521	. . . 4 ⊢ dom ℵ = On
154	153	eleq2i 2830	. . 3 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
155		ndmfv 6786	. . . 4 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
156		1n0 8286	. . . . . 6 ⊢ 1_o ≠ ∅
157		1oex 8280	. . . . . . 7 ⊢ 1_o ∈ V
158	157	0sdom 8844	. . . . . 6 ⊢ (∅ ≺ 1_o ↔ 1_o ≠ ∅)
159	156, 158	mpbir 230	. . . . 5 ⊢ ∅ ≺ 1_o
160		id 22	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
161		fveq2 6756	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(ℵ‘𝐴)) = (cf‘∅))
162		cf0 9938	. . . . . . . . 9 ⊢ (cf‘∅) = ∅
163	161, 162	eqtrdi 2795	. . . . . . . 8 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(ℵ‘𝐴)) = ∅)
164	160, 163	oveq12d 7273	. . . . . . 7 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) = (∅ ↑_m ∅))
165		0ex 5226	. . . . . . . 8 ⊢ ∅ ∈ V
166		map0e 8628	. . . . . . . 8 ⊢ (∅ ∈ V → (∅ ↑_m ∅) = 1_o)
167	165, 166	ax-mp 5	. . . . . . 7 ⊢ (∅ ↑_m ∅) = 1_o
168	164, 167	eqtrdi 2795	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) = 1_o)
169	160, 168	breq12d 5083	. . . . 5 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) ↔ ∅ ≺ 1_o))
170	159, 169	mpbiri 257	. . . 4 ⊢ ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
171	155, 170	syl 17	. . 3 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
172	154, 171	sylnbir 330	. 2 ⊢ (¬ 𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
173	151, 172	pm2.61i 182	1 ⊢ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ w3o 1084 ∧ w3a 1085 = wceq 1539 ∃wex 1783 ∈ wcel 2108 {cab 2715 ≠ wne 2942 ∀wral 3063 ∃wrex 3064 Vcvv 3422 ⊆ wss 3883 ∅c0 4253 𝒫 cpw 4530 ∪ cuni 4836 ∪ ciun 4921 class class class wbr 5070 ↦ cmpt 5153 dom cdm 5580 ran crn 5581 Oncon0 6251 Lim wlim 6252 suc csuc 6253 Fn wfn 6413 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ωcom 7687 Smo wsmo 8147 1_oc1o 8260 2_oc2o 8261 ↑_m cmap 8573 Xcixp 8643 ≈ cen 8688 ≼ cdom 8689 ≺ csdm 8690 harchar 9245 cardccrd 9624 ℵcale 9625 cfccf 9626

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-inf2 9329 ax-ac2 10150

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-se 5536 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-isom 6427 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-smo 8148 df-recs 8173 df-rdg 8212 df-1o 8267 df-2o 8268 df-er 8456 df-map 8575 df-ixp 8644 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-oi 9199 df-har 9246 df-card 9628 df-aleph 9629 df-cf 9630 df-acn 9631 df-ac 9803

This theorem is referenced by: cfpwsdom 10271 tskcard 10468 bj-pwcfsdom 35160

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