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Theorem pwcfsdom 10578

Description: A corollary of Konig's Theorem konigth 10564. 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 7835	. . . 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 10259	. . . . . . 7 ⊢ (cf‘ω) = ω
4		aleph0 10061	. . . . . . . 8 ⊢ (ℵ‘∅) = ω
5	4	fveq2i 6895	. . . . . . 7 ⊢ (cf‘(ℵ‘∅)) = (cf‘ω)
6	3, 5, 4	3eqtr4i 2771	. . . . . 6 ⊢ (cf‘(ℵ‘∅)) = (ℵ‘∅)
7		2fveq3 6897	. . . . . 6 ⊢ (𝐴 = ∅ → (cf‘(ℵ‘𝐴)) = (cf‘(ℵ‘∅)))
8		fveq2 6892	. . . . . 6 ⊢ (𝐴 = ∅ → (ℵ‘𝐴) = (ℵ‘∅))
9	6, 7, 8	3eqtr4a 2799	. . . . 5 ⊢ (𝐴 = ∅ → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
10		fvex 6905	. . . . . . . . 9 ⊢ (ℵ‘𝐴) ∈ V
11	10	canth2 9130	. . . . . . . 8 ⊢ (ℵ‘𝐴) ≺ 𝒫 (ℵ‘𝐴)
12	10	pw2en 9079	. . . . . . . 8 ⊢ 𝒫 (ℵ‘𝐴) ≈ (2_o ↑_m (ℵ‘𝐴))
13		sdomentr 9111	. . . . . . . 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 691	. . . . . . 7 ⊢ (ℵ‘𝐴) ≺ (2_o ↑_m (ℵ‘𝐴))
15		alephon 10064	. . . . . . . . 9 ⊢ (ℵ‘𝐴) ∈ On
16		alephgeom 10077	. . . . . . . . . 10 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
17		omelon 9641	. . . . . . . . . . . 12 ⊢ ω ∈ On
18		2onn 8641	. . . . . . . . . . . 12 ⊢ 2_o ∈ ω
19		onelss 6407	. . . . . . . . . . . 12 ⊢ (ω ∈ On → (2_o ∈ ω → 2_o ⊆ ω))
20	17, 18, 19	mp2 9	. . . . . . . . . . 11 ⊢ 2_o ⊆ ω
21		sstr 3991	. . . . . . . . . . 11 ⊢ ((2_o ⊆ ω ∧ ω ⊆ (ℵ‘𝐴)) → 2_o ⊆ (ℵ‘𝐴))
22	20, 21	mpan 689	. . . . . . . . . 10 ⊢ (ω ⊆ (ℵ‘𝐴) → 2_o ⊆ (ℵ‘𝐴))
23	16, 22	sylbi 216	. . . . . . . . 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 68	. . . . . . . 8 ⊢ (𝐴 ∈ On → 2_o ≼ (ℵ‘𝐴))
26		mapdom1 9142	. . . . . . . 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 9110	. . . . . . 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 588	. . . . . 6 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (ℵ‘𝐴)))
30		oveq2 7417	. . . . . . 7 ⊢ ((cf‘(ℵ‘𝐴)) = (ℵ‘𝐴) → ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) = ((ℵ‘𝐴) ↑_m (ℵ‘𝐴)))
31	30	breq2d 5161	. . . . . 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 10577	. . . . . . 7 ⊢ (cf‘(ℵ‘suc 𝑥)) = (ℵ‘suc 𝑥)
35		2fveq3 6897	. . . . . . 7 ⊢ (𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (cf‘(ℵ‘suc 𝑥)))
36		fveq2 6892	. . . . . . 7 ⊢ (𝐴 = suc 𝑥 → (ℵ‘𝐴) = (ℵ‘suc 𝑥))
37	34, 35, 36	3eqtr4a 2799	. . . . . 6 ⊢ (𝐴 = suc 𝑥 → (cf‘(ℵ‘𝐴)) = (ℵ‘𝐴))
38	37	rexlimivw 3152	. . . . 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 10266	. . . . . 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 6429	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → 𝐴 ∈ On)
42		ffn 6718	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → 𝑓 Fn (cf‘(ℵ‘𝐴)))
43		fnrnfv 6952	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 Fn (cf‘(ℵ‘𝐴)) → ran 𝑓 = {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓‘𝑥)})
44	43	unieqd 4923	. . . . . . . . . . . . . . . 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 6905	. . . . . . . . . . . . . . . 16 ⊢ (𝑓‘𝑥) ∈ V
47	46	dfiun2 5037	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) = ∪ {𝑦 ∣ ∃𝑥 ∈ (cf‘(ℵ‘𝐴))𝑦 = (𝑓‘𝑥)}
48	45, 47	eqtr4di 2791	. . . . . . . . . . . . . 14 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∪ ran 𝑓 = ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥))
49	48	ad2antrl 727	. . . . . . . . . . . . 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 7083	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓 Fn (cf‘(ℵ‘𝐴)) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑤) ∈ ran 𝑓)
51	42, 50	sylan 581	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑤) ∈ ran 𝑓)
52		sseq2 4009	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑓‘𝑤) → (𝑧 ⊆ 𝑦 ↔ 𝑧 ⊆ (𝑓‘𝑤)))
53	52	rspcev 3613	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓‘𝑤) ∈ ran 𝑓 ∧ 𝑧 ⊆ (𝑓‘𝑤)) → ∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦)
54	51, 53	sylan 581	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑤 ∈ (cf‘(ℵ‘𝐴))) ∧ 𝑧 ⊆ (𝑓‘𝑤)) → ∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦)
55	54	rexlimdva2 3158	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤) → ∃𝑦 ∈ ran 𝑓 𝑧 ⊆ 𝑦))
56	55	ralimdv 3170	. . . . . . . . . . . . . . . 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 408	. . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . 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 10078	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On ↔ Lim (ℵ‘𝐴))
60	59	biimpi 215	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ On → Lim (ℵ‘𝐴))
61		frn 6725	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ran 𝑓 ⊆ (ℵ‘𝐴))
62	61	adantr 482	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ ∀𝑧 ∈ (ℵ‘𝐴)∃𝑤 ∈ (cf‘(ℵ‘𝐴))𝑧 ⊆ (𝑓‘𝑤)) → ran 𝑓 ⊆ (ℵ‘𝐴))
63		coflim 10256	. . . . . . . . . . . . . . 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 597	. . . . . . . . . . . . . 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 257	. . . . . . . . . . . . 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 2775	. . . . . . . . . . . 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		ffvelcdm 7084	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑥) ∈ (ℵ‘𝐴))
68	15	oneli 6479	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓‘𝑥) ∈ (ℵ‘𝐴) → (𝑓‘𝑥) ∈ On)
69		harcard 9973	. . . . . . . . . . . . . . . . . . 19 ⊢ (card‘(har‘(𝑓‘𝑥))) = (har‘(𝑓‘𝑥))
70		iscard 9970	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((card‘(har‘(𝑓‘𝑥))) = (har‘(𝑓‘𝑥)) ↔ ((har‘(𝑓‘𝑥)) ∈ On ∧ ∀𝑦 ∈ (har‘(𝑓‘𝑥))𝑦 ≺ (har‘(𝑓‘𝑥))))
71	70	simprbi 498	. . . . . . . . . . . . . . . . . . 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 8983	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑓‘𝑥) ∈ V → (𝑓‘𝑥) ≼ (𝑓‘𝑥))
74	46, 73	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓‘𝑥) ≼ (𝑓‘𝑥)
75		elharval 9556	. . . . . . . . . . . . . . . . . . . 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 690	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑓‘𝑥) ∈ On → (𝑓‘𝑥) ∈ (har‘(𝑓‘𝑥)))
78		breq1 5152	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑓‘𝑥) → (𝑦 ≺ (har‘(𝑓‘𝑥)) ↔ (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
79	78	rspccv 3610	. . . . . . . . . . . . . . . . . 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 9554	. . . . . . . . . . . . . . . . . . 19 ⊢ (har‘(𝑓‘𝑥)) ∈ On
83		2fveq3 6897	. . . . . . . . . . . . . . . . . . . 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 6997	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ (cf‘(ℵ‘𝐴)) ∧ (har‘(𝑓‘𝑥)) ∈ On) → (𝐻‘𝑥) = (har‘(𝑓‘𝑥)))
86	82, 85	mpan2 690	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (𝐻‘𝑥) = (har‘(𝑓‘𝑥)))
87	86	breq2d 5161	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (cf‘(ℵ‘𝐴)) → ((𝑓‘𝑥) ≺ (𝐻‘𝑥) ↔ (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
88	87	adantl 483	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → ((𝑓‘𝑥) ≺ (𝐻‘𝑥) ↔ (𝑓‘𝑥) ≺ (har‘(𝑓‘𝑥))))
89	81, 88	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝑓‘𝑥) ≺ (𝐻‘𝑥))
90	89	ralrimiva 3147	. . . . . . . . . . . . . 14 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) ≺ (𝐻‘𝑥))
91		fvex 6905	. . . . . . . . . . . . . . 15 ⊢ (cf‘(ℵ‘𝐴)) ∈ V
92		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥) = ∪ 𝑥 ∈ (cf‘(ℵ‘𝐴))(𝑓‘𝑥)
93		eqid 2733	. . . . . . . . . . . . . . 15 ⊢ X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) = X𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥)
94	91, 92, 93	konigth 10564	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . 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 5173	. . . . . . . . . . 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 581	. . . . . . . . . 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 7442	. . . . . . . . . . . 12 ⊢ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) ∈ V
100	67	ex 414	. . . . . . . . . . . . . . . 16 ⊢ (𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (𝑓‘𝑥) ∈ (ℵ‘𝐴)))
101		alephlim 10062	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → (ℵ‘𝐴) = ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦))
102	101	eleq2d 2820	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∈ V ∧ Lim 𝐴) → ((𝑓‘𝑥) ∈ (ℵ‘𝐴) ↔ (𝑓‘𝑥) ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦)))
103		eliun 5002	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑓‘𝑥) ∈ ∪ 𝑦 ∈ 𝐴 (ℵ‘𝑦) ↔ ∃𝑦 ∈ 𝐴 (𝑓‘𝑥) ∈ (ℵ‘𝑦))
104		alephcard 10065	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (card‘(ℵ‘𝑦)) = (ℵ‘𝑦)
105	104	eleq2i 2826	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓‘𝑥) ∈ (card‘(ℵ‘𝑦)) ↔ (𝑓‘𝑥) ∈ (ℵ‘𝑦))
106		cardsdomelir 9968	. . . . . . . . . . . . . . . . . . . . . . 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 9556	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)) ↔ ((ℵ‘𝑦) ∈ On ∧ (ℵ‘𝑦) ≼ (𝑓‘𝑥)))
109	108	simprbi 498	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥)) → (ℵ‘𝑦) ≼ (𝑓‘𝑥))
110		domnsym 9099	. . . . . . . . . . . . . . . . . . . . . . . . 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 10064	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (ℵ‘𝑦) ∈ On
114		ontri1 6399	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((har‘(𝑓‘𝑥)) ∈ On ∧ (ℵ‘𝑦) ∈ On) → ((har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦) ↔ ¬ (ℵ‘𝑦) ∈ (har‘(𝑓‘𝑥))))
115	82, 113, 114	mp2an 691	. . . . . . . . . . . . . . . . . . . . . . 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 10072	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐴 ∈ On → (𝑦 ∈ 𝐴 → (ℵ‘𝑦) ∈ (ℵ‘𝐴)))
119	118	imp 408	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) → (ℵ‘𝑦) ∈ (ℵ‘𝐴))
120		ontr2 6412	. . . . . . . . . . . . . . . . . . . . . 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 691	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((har‘(𝑓‘𝑥)) ⊆ (ℵ‘𝑦) ∧ (ℵ‘𝑦) ∈ (ℵ‘𝐴)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴))
122	117, 119, 121	syl2anr 598	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐴 ∈ On ∧ 𝑦 ∈ 𝐴) ∧ (𝑓‘𝑥) ∈ (ℵ‘𝑦)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴))
123	122	rexlimdva2 3158	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∈ On → (∃𝑦 ∈ 𝐴 (𝑓‘𝑥) ∈ (ℵ‘𝑦) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
124	103, 123	biimtrid 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 510	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → (𝑥 ∈ (cf‘(ℵ‘𝐴)) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴)))
128	127	imp 408	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (har‘(𝑓‘𝑥)) ∈ (ℵ‘𝐴))
129	83	cbvmptv 5262	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑦))) = (𝑥 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑥)))
130	84, 129	eqtri 2761	. . . . . . . . . . . . . 14 ⊢ 𝐻 = (𝑥 ∈ (cf‘(ℵ‘𝐴)) ↦ (har‘(𝑓‘𝑥)))
131	128, 130	fmptd 7114	. . . . . . . . . . . . 13 ⊢ (((𝐴 ∈ V ∧ Lim 𝐴) ∧ 𝑓:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴)) → 𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴))
132		ffvelcdm 7084	. . . . . . . . . . . . . . 15 ⊢ ((𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) ∧ 𝑥 ∈ (cf‘(ℵ‘𝐴))) → (𝐻‘𝑥) ∈ (ℵ‘𝐴))
133		onelss 6407	. . . . . . . . . . . . . . 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 3147	. . . . . . . . . . . . 13 ⊢ (𝐻:(cf‘(ℵ‘𝐴))⟶(ℵ‘𝐴) → ∀𝑥 ∈ (cf‘(ℵ‘𝐴))(𝐻‘𝑥) ⊆ (ℵ‘𝐴))
136		ss2ixp 8904	. . . . . . . . . . . . . 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 8901	. . . . . . . . . . . . . 14 ⊢ X𝑥 ∈ (cf‘(ℵ‘𝐴))(ℵ‘𝐴) = ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴)))
138	136, 137	sseqtrdi 4033	. . . . . . . . . . . . 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 8996	. . . . . . . . . . . 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 716	. . . . . . . . . 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 9110	. . . . . . . . . 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 585	. . . . . . . . 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 415	. . . . . . . 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 1132	. . . . . . 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 1429	. . 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 10060	. . . . 5 ⊢ ℵ Fn On
153	152	fndmi 6654	. . . 4 ⊢ dom ℵ = On
154	153	eleq2i 2826	. . 3 ⊢ (𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
155		ndmfv 6927	. . . 4 ⊢ (¬ 𝐴 ∈ dom ℵ → (ℵ‘𝐴) = ∅)
156		1n0 8488	. . . . . 6 ⊢ 1_o ≠ ∅
157		1oex 8476	. . . . . . 7 ⊢ 1_o ∈ V
158	157	0sdom 9107	. . . . . 6 ⊢ (∅ ≺ 1_o ↔ 1_o ≠ ∅)
159	156, 158	mpbir 230	. . . . 5 ⊢ ∅ ≺ 1_o
160		id 22	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → (ℵ‘𝐴) = ∅)
161		fveq2 6892	. . . . . . . . 9 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(ℵ‘𝐴)) = (cf‘∅))
162		cf0 10246	. . . . . . . . 9 ⊢ (cf‘∅) = ∅
163	161, 162	eqtrdi 2789	. . . . . . . 8 ⊢ ((ℵ‘𝐴) = ∅ → (cf‘(ℵ‘𝐴)) = ∅)
164	160, 163	oveq12d 7427	. . . . . . 7 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) = (∅ ↑_m ∅))
165		0ex 5308	. . . . . . . 8 ⊢ ∅ ∈ V
166		map0e 8876	. . . . . . . 8 ⊢ (∅ ∈ V → (∅ ↑_m ∅) = 1_o)
167	165, 166	ax-mp 5	. . . . . . 7 ⊢ (∅ ↑_m ∅) = 1_o
168	164, 167	eqtrdi 2789	. . . . . 6 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) = 1_o)
169	160, 168	breq12d 5162	. . . . 5 ⊢ ((ℵ‘𝐴) = ∅ → ((ℵ‘𝐴) ≺ ((ℵ‘𝐴) ↑_m (cf‘(ℵ‘𝐴))) ↔ ∅ ≺ 1_o))
170	159, 169	mpbiri 258	. . . 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 331	. 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 397 ∨ w3o 1087 ∧ w3a 1088 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {cab 2710 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 Vcvv 3475 ⊆ wss 3949 ∅c0 4323 𝒫 cpw 4603 ∪ cuni 4909 ∪ ciun 4998 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5677 ran crn 5678 Oncon0 6365 Lim wlim 6366 suc csuc 6367 Fn wfn 6539 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 ωcom 7855 Smo wsmo 8345 1_oc1o 8459 2_oc2o 8460 ↑_m cmap 8820 Xcixp 8891 ≈ cen 8936 ≼ cdom 8937 ≺ csdm 8938 harchar 9551 cardccrd 9930 ℵcale 9931 cfccf 9932

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-ac2 10458

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-smo 8346 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-er 8703 df-map 8822 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-oi 9505 df-har 9552 df-card 9934 df-aleph 9935 df-cf 9936 df-acn 9937 df-ac 10111

This theorem is referenced by: cfpwsdom 10579 tskcard 10776 bj-pwcfsdom 35943

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