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Theorem pwcfsdom 10567

Description: A corollary of Konig's Theorem konigth 10553. 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 7841	. . . 4 ⊢ (𝐴 ∈ On ↔ (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
2	1	biimpi 219	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = ∅ ∨ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ∨ (𝐴 ∈ V ∧ Lim 𝐴)))
3		cfom 10247	. . . . . . 7 ⊢ (cf‘ω) = ω
4		aleph0 10049	. . . . . . . 8 ⊢ (ℵ‘∅) = ω
5	4	fveq2i 6884	. . . . . . 7 ⊢ (cf‘(ℵ‘∅)) = (cf‘ω)
6	3, 5, 4	3eqtr4i 2794	. . . . . 6 ⊢ (cf‘(ℵ‘∅)) = (ℵ‘∅)
7		2fveq3 6886	. . . . . 6 ⊢ (𝐴 = ∅ → (cf‘(ℵ‘𝐴)) = (cf‘(ℵ‘∅)))
8		fveq2 6881	. . . . . 6 ⊢ (𝐴 = ∅ → (ℵ‘𝐴) = (ℵ‘∅))
9	6, 7, 8	3eqtr4a 2822	. . . . 5 ⊢ (𝐴 = ∅ → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
10		fvex 6894	. . . . . . . . 9 ⊢ (ℵ‘𝐴) ∈ V
11	10	canth2 9117	. . . . . . . 8 ⊢ (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
12	10	pw2en 9071	. . . . . . . 8 ⊢ 𝒫 (ℵ‘𝐴) ≈ (2_o ↑_m (ℵ‘𝐴))
13		sdomentr 9098	. . . . . . . 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 704	. . . . . . 7 ⊢ (ℵ‘𝐴) ≺ (2_o ↑_m (ℵ‘𝐴))
15		alephon 10052	. . . . . . . . 9 ⊢ (ℵ‘𝐴) ∈ On
16		alephgeom 10065	. . . . . . . . . 10 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
17		omelon 9614	. . . . . . . . . . . 12 ⊢ ω ∈ On
18		2onn 8627	. . . . . . . . . . . 12 ⊢ 2_o ∈ ω
19		onelss 6403	. . . . . . . . . . . 12 ⊢ (ω ∈ On → (2_o ∈ ω → 2_o ⊆ ω))
20	17, 18, 19	mp2 9	. . . . . . . . . . 11 ⊢ 2_o ⊆ ω
21		sstr 3944	. . . . . . . . . . 11 ⊢ ((2_o ⊆ ω ∧ ω ⊆ (ℵ‘𝐴)) → 2_o ⊆ (ℵ‘𝐴))
22	20, 21	mpan 702	. . . . . . . . . 10 ⊢ (ω ⊆ (ℵ‘𝐴) → 2_o ⊆ (ℵ‘𝐴))
23	16, 22	sylbi 220	. . . . . . . . 9 ⊢ (𝐴 ∈ On → 2_o ⊆ (ℵ‘𝐴))
24		ssdomg 8996	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) ∈ On → (2_o ⊆ (ℵ‘𝐴) → 2_o ≼ (ℵ‘𝐴)))
25	15, 23, 24	mpsyl 69	. . . . . . . 8 ⊢ (𝐴 ∈ On → 2_o ≼ (ℵ‘𝐴))
26		mapdom1 9129	. . . . . . . 8 ⊢ (2_o ≼ (ℵ‘𝐴) → (2_o ↑_m (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑_m (ℵ‘𝐴)))
27	25, 26	syl 18	. . . . . . 7 ⊢ (𝐴 ∈ On → (2_o ↑_m (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) ↑_m (ℵ‘𝐴)))
28		sdomdomtr 9097	. . . . . . 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 598	. . . . . 6 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (ℵ‘𝐴)))
30		oveq2 7418	. . . . . . 7 ⊢ ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) = ((ℵ‘𝐴) ↑_m (ℵ‘𝐴)))
31	30	breq2d 5120	. . . . . 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 250	. . . . 5 ⊢ (𝐴 ∈ On → ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))))
33	9, 32	syl5 35	. . . 4 ⊢ (𝐴 ∈ On → (𝐴 = ∅ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))))
34		alephreg 10566	. . . . . . 7 ⊢ (cf‘(ℵ‘suc 𝑥)) = (ℵ‘suc 𝑥)
35		2fveq3 6886	. . . . . . 7 ⊢ (𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (cf‘(ℵ‘suc 𝑥)))
36		fveq2 6881	. . . . . . 7 ⊢ (𝐴 = suc 𝑥 → (ℵ‘𝐴) = (ℵ‘suc 𝑥))
37	34, 35, 36	3eqtr4a 2822	. . . . . 6 ⊢ (𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
38	37	rexlimivw 3160	. . . . 5 ⊢ (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
39	38, 32	syl5 35	. . . 4 ⊢ (𝐴 ∈ On → (∃𝑥 ∈ On 𝐴 = suc 𝑥 → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))))
40		limelon 6426	. . . . . . . . . 10 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
41		ffn 6705	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → 𝑓 Fn (cf‘(ℵ‘𝐴)))
42		fnrnfv 6940	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 Fn (cf‘(ℵ‘𝐴)) → ran 𝑓 = {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓‘𝑥)})
43	42	unieqd 4884	. . . . . . . . . . . . . . 15 ⊢ (𝑓 Fn (cf‘(ℵ‘𝐴)) → ∪ ran 𝑓 = ∪ {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓‘𝑥)})
44	41, 43	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∪ ran 𝑓 = ∪ {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓‘𝑥)})
45		fvex 6894	. . . . . . . . . . . . . . 15 ⊢ (𝑓‘𝑥) ∈ V
46	45	dfiun2 4995	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) = ∪ {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓‘𝑥)}
47	44, 46	eqtr4di 2814	. . . . . . . . . . . . 13 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∪ ran 𝑓 = ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥))
48	47	ad2antrl 740	. . . . . . . . . . . 12 ⊢ ((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))
49		fnfvelrn 7075	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓 Fn (cf‘(ℵ‘𝐴)) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑤) ∈ ran 𝑓)
50	41, 49	sylan 591	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑤) ∈ ran 𝑓)
51		sseq2 3962	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑓‘𝑤) → (𝑧 ⊆ 𝑦 ↔ 𝑧 ⊆ (𝑓‘𝑤)))
52	51	rspcev 3580	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓‘𝑤) ∈ ran 𝑓 ∧ 𝑧 ⊆ (𝑓‘𝑤)) → ∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦)
53	50, 52	sylan 591	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) ∧ 𝑧 ⊆ (𝑓‘𝑤)) → ∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦)
54	53	rexlimdva2 3166	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤) → ∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦))
55	54	ralimdv 3177	. . . . . . . . . . . . . . 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))
56	55	imp 411	. . . . . . . . . . . . . 14 ⊢ ((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	adantl 486	. . . . . . . . . . . . 13 ⊢ ((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)
58		alephislim 10066	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ On ↔ Lim (ℵ‘𝐴))
59	58	biimpi 219	. . . . . . . . . . . . . 14 ⊢ (𝐴 ∈ On → Lim (ℵ‘𝐴))
60		frn 6713	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ran 𝑓 ⊆ (ℵ‘𝐴))
61	60	adantr 485	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤)) → ran 𝑓 ⊆ (ℵ‘𝐴))
62		coflim 10244	. . . . . . . . . . . . . 14 ⊢ ((Lim (ℵ‘𝐴) ∧ ran 𝑓 ⊆ (ℵ‘𝐴)) → (∪ ran 𝑓 = (ℵ‘𝐴) ↔ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦))
63	59, 61, 62	syl2an 607	. . . . . . . . . . . . 13 ⊢ ((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))
64	57, 63	mpbird 260	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))) → ∪ ran 𝑓 = (ℵ‘𝐴))
65	48, 64	eqtr3d 2798	. . . . . . . . . . 11 ⊢ ((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))
66		ffvelcdm 7076	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑥) ∈ (ℵ‘𝐴))
67	15	oneli 6476	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑥) ∈ (ℵ‘𝐴) → (𝑓‘𝑥) ∈ On)
68		harcard 9963	. . . . . . . . . . . . . . . . . 18 ⊢ (card‘(har‘(𝑓‘𝑥))) = (har‘(𝑓‘𝑥))
69		iscard 9960	. . . . . . . . . . . . . . . . . . 19 ⊢ ((card‘(har‘(𝑓‘𝑥))) = (har‘(𝑓‘𝑥)) ↔ ((har‘(𝑓‘𝑥)) ∈ On ∧ ∀𝑦 ∈ (har‘(𝑓‘𝑥))𝑦 ≺ (har‘(𝑓‘𝑥))))
70	69	simprbi 502	. . . . . . . . . . . . . . . . . 18 ⊢ ((card‘(har‘(𝑓‘𝑥))) = (har‘(𝑓‘𝑥)) → ∀𝑦 ∈ (har‘(𝑓‘𝑥))𝑦 ≺ (har‘(𝑓‘𝑥)))
71	68, 70	ax-mp 5	. . . . . . . . . . . . . . . . 17 ⊢ ∀𝑦 ∈ (har‘(𝑓‘𝑥))𝑦 ≺ (har‘(𝑓‘𝑥))
72		domrefg 8983	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑥) ∈ V → (𝑓‘𝑥) ≼ (𝑓‘𝑥))
73	45, 72	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓‘𝑥) ≼ (𝑓‘𝑥)
74		elharval 9522	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑥) ∈ (har‘(𝑓‘𝑥)) ↔ ((𝑓‘𝑥) ∈ On ∧ (𝑓‘𝑥) ≼ (𝑓‘𝑥)))
75	74	biimpri 231	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓‘𝑥) ∈ On ∧ (𝑓‘𝑥) ≼ (𝑓‘𝑥)) → (𝑓‘𝑥) ∈ (har‘(𝑓‘𝑥)))
76	73, 75	mpan2 703	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑥) ∈ On → (𝑓‘𝑥) ∈ (har‘(𝑓‘𝑥)))
77		breq1 5111	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑓‘𝑥) → (𝑦 ≺ (har‘(𝑓‘𝑥)) ↔ (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
78	77	rspccv 3577	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑦 ∈ (har‘(𝑓‘𝑥))𝑦 ≺ (har‘(𝑓‘𝑥)) → ((𝑓‘𝑥) ∈ (har‘(𝑓‘𝑥)) → (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
79	71, 76, 78	mpsyl 69	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓‘𝑥) ∈ On → (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥)))
80	66, 67, 79	3syl 19	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥)))
81		harcl 9520	. . . . . . . . . . . . . . . . . 18 ⊢ (har‘(𝑓‘𝑥)) ∈ On
82		2fveq3 6886	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑥 → (har‘(𝑓‘𝑦)) = (har‘(𝑓‘𝑥)))
83		pwcfsdom.1	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐻 = (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑦)))
84	82, 83	fvmptg 6987	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ (cf‘(ℵ‘𝐴)) ∧ (har‘(𝑓‘𝑥)) ∈ On) → (𝐻‘𝑥) = (har‘(𝑓‘𝑥)))
85	81, 84	mpan2 703	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (𝐻‘𝑥) = (har‘(𝑓‘𝑥)))
86	85	breq2d 5120	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (cf‘(ℵ‘𝐴)) → ((𝑓‘𝑥) ≺ (𝐻‘𝑥) ↔ (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
87	86	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → ((𝑓‘𝑥) ≺ (𝐻‘𝑥) ↔ (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
88	80, 87	mpbird 260	. . . . . . . . . . . . . 14 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑥) ≺ (𝐻‘𝑥))
89	88	ralrimiva 3155	. . . . . . . . . . . . 13 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ (𝐻‘𝑥))
90		fvex 6894	. . . . . . . . . . . . . 14 ⊢ (cf‘(ℵ‘𝐴)) ∈ V
91		eqid 2761	. . . . . . . . . . . . . 14 ⊢ ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) = ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥)
92		eqid 2761	. . . . . . . . . . . . . 14 ⊢ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) = X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥)
93	90, 91, 92	konigth 10553	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ (𝐻‘𝑥) → ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥))
94	89, 93	syl 18	. . . . . . . . . . . 12 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥))
95	94	ad2antrl 740	. . . . . . . . . . 11 ⊢ ((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))
96	65, 95	eqbrtrrd 5134	. . . . . . . . . 10 ⊢ ((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))
97	40, 96	sylan 591	. . . . . . . . 9 ⊢ (((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))
98		ovex 7443	. . . . . . . . . . 11 ⊢ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) ∈ V
99	66	ex 417	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (𝑓‘𝑥) ∈ (ℵ‘𝐴)))
100		alephlim 10050	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) = ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦))
101	100	eleq2d 2847	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓‘𝑥) ∈ (ℵ‘𝐴) ↔ (𝑓‘𝑥) ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦)))
102		eliun 4959	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑥) ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) ↔ ∃𝑦 ∈ 𝐴 (𝑓‘𝑥) ∈ (ℵ‘𝑦))
103		alephcard 10053	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)
104	103	eleq2i 2853	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓‘𝑥) ∈ (card‘(ℵ‘𝑦)) ↔ (𝑓‘𝑥) ∈ (ℵ‘𝑦))
105		cardsdomelir 9958	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓‘𝑥) ∈ (card‘(ℵ‘𝑦)) → (𝑓‘𝑥) ≺ (ℵ‘𝑦))
106	104, 105	sylbir 238	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓‘𝑥) ∈ (ℵ‘𝑦) → (𝑓‘𝑥) ≺ (ℵ‘𝑦))
107		elharval 9522	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)) ↔ ((ℵ‘𝑦) ∈ On ∧ (ℵ‘𝑦) ≼ (𝑓‘𝑥)))
108	107	simprbi 502	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)) → (ℵ‘𝑦) ≼ (𝑓‘𝑥))
109		domnsym 9090	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ℵ‘𝑦) ≼ (𝑓‘𝑥) → ¬ (𝑓‘𝑥) ≺ (ℵ‘𝑦))
110	108, 109	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)) → ¬ (𝑓‘𝑥) ≺ (ℵ‘𝑦))
111	110	con2i 140	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓‘𝑥) ≺ (ℵ‘𝑦) → ¬ (ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)))
112		alephon 10052	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (ℵ‘𝑦) ∈ On
113		ontri1 6395	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((har‘(𝑓‘𝑥)) ∈ On ∧ (ℵ‘𝑦) ∈ On) → ((har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦) ↔ ¬ (ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥))))
114	81, 112, 113	mp2an 704	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦) ↔ ¬ (ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)))
115	111, 114	sylibr 237	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑓‘𝑥) ≺ (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦))
116	106, 115	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦))
117		alephord2i 10060	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
118	117	imp 411	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → (ℵ‘𝑦) ∈ (ℵ‘𝐴))
119		ontr2 6409	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((har‘(𝑓‘𝑥)) ∈ On ∧ (ℵ‘𝐴) ∈ On) → (((har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝐴)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
120	81, 15, 119	mp2an 704	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝐴)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴))
121	116, 118, 120	syl2anr 608	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑓‘𝑥) ∈ (ℵ‘𝑦)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴))
122	121	rexlimdva2 3166	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ On → (∃𝑦 ∈ 𝐴 (𝑓‘𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
123	102, 122	biimtrid 245	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ On → ((𝑓‘𝑥) ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
124	40, 123	syl 18	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓‘𝑥) ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
125	101, 124	sylbid 243	. . . . . . . . . . . . . . 15 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓‘𝑥) ∈ (ℵ‘𝐴) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
126	99, 125	sylan9r 517	. . . . . . . . . . . . . 14 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
127	126	imp 411	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴))
128	82	cbvmptv 5214	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑦))) = (𝑥 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑥)))
129	83, 128	eqtri 2784	. . . . . . . . . . . . 13 ⊢ 𝐻 = (𝑥 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑥)))
130	127, 129	fmptd 7109	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → 𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴))
131		ffvelcdm 7076	. . . . . . . . . . . . . 14 ⊢ ((𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝐻‘𝑥) ∈ (ℵ‘𝐴))
132		onelss 6403	. . . . . . . . . . . . . 14 ⊢ ((ℵ‘𝐴) ∈ On → ((𝐻‘𝑥) ∈ (ℵ‘𝐴) → (𝐻‘𝑥) ⊆ (ℵ‘𝐴)))
133	15, 131, 132	mpsyl 69	. . . . . . . . . . . . 13 ⊢ ((𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝐻‘𝑥) ⊆ (ℵ‘𝐴))
134	133	ralrimiva 3155	. . . . . . . . . . . 12 ⊢ (𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ (ℵ‘𝐴))
135		ss2ixp 8907	. . . . . . . . . . . . 13 ⊢ (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ (ℵ‘𝐴) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ X𝑥 ∈ (cf‘(ℵ‘𝐴))(ℵ‘𝐴))
136	90, 10	ixpconst 8904	. . . . . . . . . . . . 13 ⊢ X𝑥 ∈ (cf‘(ℵ‘𝐴))(ℵ‘𝐴) = ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))
137	135, 136	sseqtrdi 3976	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ (ℵ‘𝐴) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
138	130, 134, 137	3syl 19	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
139		ssdomg 8996	. . . . . . . . . . 11 ⊢ (((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)))))
140	98, 138, 139	mpsyl 69	. . . . . . . . . 10 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ≼ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
141	140	adantrr 729	. . . . . . . . 9 ⊢ (((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))))
142		sdomdomtr 9097	. . . . . . . . 9 ⊢ (((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))))
143	97, 141, 142	syl2anc 595	. . . . . . . 8 ⊢ (((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))))
144	143	expcom 418	. . . . . . 7 ⊢ ((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)))))
145	144	3adant2 1147	. . . . . 6 ⊢ ((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)))))
146		cfsmo 10254	. . . . . . 7 ⊢ ((ℵ‘𝐴) ∈ On → ∃𝑓(𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤)))
147	15, 146	ax-mp 5	. . . . . 6 ⊢ ∃𝑓(𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ Smo 𝑓 ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤))
148	145, 147	exlimiiv 1959	. . . . 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 1454	. . 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 16	. 2 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
152		alephfnon 10048	. . . . 5 ⊢ ℵ Fn On
153	152	fndmi 6639	. . . 4 ⊢ dom ℵ = On
154	153	eleq2i 2853	. . 3 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
155		ndmfv 6913	. . . 4 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
156		1n0 8471	. . . . . 6 ⊢ 1_o ≠ ∅
157		1oex 8462	. . . . . . 7 ⊢ 1_o ∈ V
158	157	0sdom 9095	. . . . . 6 ⊢ (∅ ≺ 1_o ↔ 1_o ≠ ∅)
159	156, 158	mpbir 234	. . . . 5 ⊢ ∅ ≺ 1_o
160		id 23	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
161		fveq2 6881	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(ℵ‘𝐴)) = (cf‘∅))
162		cf0 10233	. . . . . . . . 9 ⊢ (cf‘∅) = ∅
163	161, 162	eqtrdi 2812	. . . . . . . 8 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(ℵ‘𝐴)) = ∅)
164	160, 163	oveq12d 7428	. . . . . . 7 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) = (∅ ↑_m ∅))
165		0ex 5269	. . . . . . . 8 ⊢ ∅ ∈ V
166		map0e 8879	. . . . . . . 8 ⊢ (∅ ∈ V → (∅ ↑_m ∅) = 1_o)
167	165, 166	ax-mp 5	. . . . . . 7 ⊢ (∅ ↑_m ∅) = 1_o
168	164, 167	eqtrdi 2812	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) = 1_o)
169	160, 168	breq12d 5121	. . . . 5 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) ↔ ∅ ≺ 1_o))
170	159, 169	mpbiri 261	. . . 4 ⊢ ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
171	155, 170	syl 18	. . 3 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
172	154, 171	sylnbir 334	. 2 ⊢ (¬ 𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))))
173	151, 172	pm2.61i 184	1 ⊢ (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 ∨ w3o 1100 ∧ w3a 1101 = wceq 1568 ∃wex 1807 ∈ wcel 2141 {cab 2739 ≠ wne 2956 ∀wral 3077 ∃wrex 3087 Vcvv 3453 ⊆ wss 3904 ∅c0 4285 𝒫 cpw 4561 ∪ cuni 4871 ∪ ciun 4955 class class class wbr 5108 ↦ cmpt 5191 dom cdm 5661 ran crn 5662 Oncon0 6360 Lim wlim 6361 suc csuc 6362 Fn wfn 6531 ⟶wf 6532 ‘cfv 6536 (class class class)co 7410 ωcom 7861 Smo wsmo 8331 1_oc1o 8445 2_oc2o 8446 ↑_m cmap 8823 Xcixp 8894 ≈ cen 8939 ≼ cdom 8940 ≺ csdm 8941 harchar 9517 cardccrd 9920 ℵcale 9921 cfccf 9922

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1823 ax-4 1837 ax-5 1938 ax-6 1995 ax-7 2036 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5237 ax-sep 5256 ax-nul 5268 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9609 ax-ac2 10446

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1571 df-fal 1581 df-ex 1808 df-nf 1812 df-sb 2095 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3078 df-rex 3088 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-int 4912 df-iun 4957 df-iin 4958 df-br 5109 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-om 7862 df-1st 7985 df-2nd 7986 df-frecs 8277 df-wrecs 8308 df-smo 8332 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-er 8693 df-map 8825 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-oi 9471 df-har 9518 df-card 9924 df-aleph 9925 df-cf 9926 df-acn 9927 df-ac 10099

This theorem is referenced by: cfpwsdom 10568 tskcard 10765 bj-pwcfsdom 37642

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