Theorem alephreg 10651

Description: A successor aleph is regular. Theorem 11.15 of [TakeutiZaring] p. 103. (Contributed by Mario Carneiro, 9-Mar-2013.)

Assertion

Ref	Expression
alephreg	⊢ (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)

Proof of Theorem alephreg

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

Step	Hyp	Ref	Expression
1		alephordilem1 10142	. . . 4 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
2		alephon 10138	. . . . . . . . 9 ⊢ (ℵ‘suc 𝐴) ∈ On
3		cff1 10327	. . . . . . . . 9 ⊢ ((ℵ‘suc 𝐴) ∈ On → ∃𝑓(𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦)))
4	2, 3	ax-mp 5	. . . . . . . 8 ⊢ ∃𝑓(𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦))
5		fvex 6933	. . . . . . . . . . . . 13 ⊢ (cf‘(ℵ‘suc 𝐴)) ∈ V
6		fvex 6933	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑦) ∈ V
7	6	sucex 7842	. . . . . . . . . . . . 13 ⊢ suc (𝑓‘𝑦) ∈ V
8	5, 7	iunex 8009	. . . . . . . . . . . 12 ⊢ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) ∈ V
9		f1f 6817	. . . . . . . . . . . . . 14 ⊢ (𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) → 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴))
10	9	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))) → 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴))
11		simplr 768	. . . . . . . . . . . . 13 ⊢ (((𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))) → ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦))
12	2	oneli 6509	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (ℵ‘suc 𝐴) → 𝑥 ∈ On)
13		ffvelcdm 7115	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴))
14		onelon 6420	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((ℵ‘suc 𝐴) ∈ On ∧ (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴)) → (𝑓‘𝑦) ∈ On)
15	2, 13, 14	sylancr 586	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑓‘𝑦) ∈ On)
16		onsssuc 6485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ On ∧ (𝑓‘𝑦) ∈ On) → (𝑥 ⊆ (𝑓‘𝑦) ↔ 𝑥 ∈ suc (𝑓‘𝑦)))
17	15, 16	sylan2 592	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 ∈ On ∧ (𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴)))) → (𝑥 ⊆ (𝑓‘𝑦) ↔ 𝑥 ∈ suc (𝑓‘𝑦)))
18	17	anassrs 467	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑥 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑥 ⊆ (𝑓‘𝑦) ↔ 𝑥 ∈ suc (𝑓‘𝑦)))
19	18	rexbidva 3183	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑥 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → (∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦) ↔ ∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ∈ suc (𝑓‘𝑦)))
20		eliun 5019	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) ↔ ∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ∈ suc (𝑓‘𝑦))
21	19, 20	bitr4di 289	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑥 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → (∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦) ↔ 𝑥 ∈ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦)))
22	21	ancoms 458	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑥 ∈ On) → (∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦) ↔ 𝑥 ∈ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦)))
23	12, 22	sylan2 592	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑥 ∈ (ℵ‘suc 𝐴)) → (∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦) ↔ 𝑥 ∈ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦)))
24	23	ralbidva 3182	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) → (∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦) ↔ ∀𝑥 ∈ (ℵ‘suc 𝐴)𝑥 ∈ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦)))
25		dfss3 3997	. . . . . . . . . . . . . . 15 ⊢ ((ℵ‘suc 𝐴) ⊆ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) ↔ ∀𝑥 ∈ (ℵ‘suc 𝐴)𝑥 ∈ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦))
26	24, 25	bitr4di 289	. . . . . . . . . . . . . 14 ⊢ (𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) → (∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦) ↔ (ℵ‘suc 𝐴) ⊆ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦)))
27	26	biimpa 476	. . . . . . . . . . . . 13 ⊢ ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦)) → (ℵ‘suc 𝐴) ⊆ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦))
28	10, 11, 27	syl2anc 583	. . . . . . . . . . . 12 ⊢ (((𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))) → (ℵ‘suc 𝐴) ⊆ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦))
29		ssdomg 9060	. . . . . . . . . . . 12 ⊢ (∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) ∈ V → ((ℵ‘suc 𝐴) ⊆ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) → (ℵ‘suc 𝐴) ≼ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦)))
30	8, 28, 29	mpsyl 68	. . . . . . . . . . 11 ⊢ (((𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))) → (ℵ‘suc 𝐴) ≼ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦))
31		simprl 770	. . . . . . . . . . . 12 ⊢ (((𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))) → 𝐴 ∈ On)
32		onsuc 7847	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
33		alephislim 10152	. . . . . . . . . . . . . . . . . . 19 ⊢ (suc 𝐴 ∈ On ↔ Lim (ℵ‘suc 𝐴))
34		limsuc 7886	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim (ℵ‘suc 𝐴) → ((𝑓‘𝑦) ∈ (ℵ‘suc 𝐴) ↔ suc (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴)))
35	33, 34	sylbi 217	. . . . . . . . . . . . . . . . . 18 ⊢ (suc 𝐴 ∈ On → ((𝑓‘𝑦) ∈ (ℵ‘suc 𝐴) ↔ suc (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴)))
36	32, 35	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ On → ((𝑓‘𝑦) ∈ (ℵ‘suc 𝐴) ↔ suc (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴)))
37		breq1 5169	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = suc (𝑓‘𝑦) → (𝑧 ≺ (ℵ‘suc 𝐴) ↔ suc (𝑓‘𝑦) ≺ (ℵ‘suc 𝐴)))
38		alephcard 10139	. . . . . . . . . . . . . . . . . . . 20 ⊢ (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
39		iscard 10044	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴) ↔ ((ℵ‘suc 𝐴) ∈ On ∧ ∀𝑧 ∈ (ℵ‘suc 𝐴)𝑧 ≺ (ℵ‘suc 𝐴)))
40	39	simprbi 496	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴) → ∀𝑧 ∈ (ℵ‘suc 𝐴)𝑧 ≺ (ℵ‘suc 𝐴))
41	38, 40	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ ∀𝑧 ∈ (ℵ‘suc 𝐴)𝑧 ≺ (ℵ‘suc 𝐴)
42	37, 41	vtoclri 3603	. . . . . . . . . . . . . . . . . 18 ⊢ (suc (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓‘𝑦) ≺ (ℵ‘suc 𝐴))
43		alephsucdom 10148	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ On → (suc (𝑓‘𝑦) ≼ (ℵ‘𝐴) ↔ suc (𝑓‘𝑦) ≺ (ℵ‘suc 𝐴)))
44	42, 43	imbitrrid 246	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ On → (suc (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓‘𝑦) ≼ (ℵ‘𝐴)))
45	36, 44	sylbid 240	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On → ((𝑓‘𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓‘𝑦) ≼ (ℵ‘𝐴)))
46	13, 45	syl5 34	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ On → ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → suc (𝑓‘𝑦) ≼ (ℵ‘𝐴)))
47	46	expdimp 452	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → (𝑦 ∈ (cf‘(ℵ‘suc 𝐴)) → suc (𝑓‘𝑦) ≼ (ℵ‘𝐴)))
48	47	ralrimiv 3151	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → ∀𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) ≼ (ℵ‘𝐴))
49		iundom 10611	. . . . . . . . . . . . 13 ⊢ (((cf‘(ℵ‘suc 𝐴)) ∈ V ∧ ∀𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) ≼ (ℵ‘𝐴)) → ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)))
50	5, 48, 49	sylancr 586	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)))
51	31, 10, 50	syl2anc 583	. . . . . . . . . . 11 ⊢ (((𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))) → ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)))
52		domtr 9067	. . . . . . . . . . 11 ⊢ (((ℵ‘suc 𝐴) ≼ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) ∧ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴))) → (ℵ‘suc 𝐴) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)))
53	30, 51, 52	syl2anc 583	. . . . . . . . . 10 ⊢ (((𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))) → (ℵ‘suc 𝐴) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)))
54	53	expcom 413	. . . . . . . . 9 ⊢ ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → ((𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦)) → (ℵ‘suc 𝐴) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴))))
55	54	exlimdv 1932	. . . . . . . 8 ⊢ ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → (∃𝑓(𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓‘𝑦)) → (ℵ‘suc 𝐴) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴))))
56	4, 55	mpi 20	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → (ℵ‘suc 𝐴) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)))
57		alephgeom 10151	. . . . . . . . . 10 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
58		alephon 10138	. . . . . . . . . . 11 ⊢ (ℵ‘𝐴) ∈ On
59		infxpen 10083	. . . . . . . . . . 11 ⊢ (((ℵ‘𝐴) ∈ On ∧ ω ⊆ (ℵ‘𝐴)) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
60	58, 59	mpan 689	. . . . . . . . . 10 ⊢ (ω ⊆ (ℵ‘𝐴) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
61	57, 60	sylbi 217	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
62		breq1 5169	. . . . . . . . . . . 12 ⊢ (𝑧 = (cf‘(ℵ‘suc 𝐴)) → (𝑧 ≺ (ℵ‘suc 𝐴) ↔ (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴)))
63	62, 41	vtoclri 3603	. . . . . . . . . . 11 ⊢ ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴))
64		alephsucdom 10148	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ≼ (ℵ‘𝐴) ↔ (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴)))
65	63, 64	imbitrrid 246	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) ≼ (ℵ‘𝐴)))
66		fvex 6933	. . . . . . . . . . 11 ⊢ (ℵ‘𝐴) ∈ V
67	66	xpdom1 9137	. . . . . . . . . 10 ⊢ ((cf‘(ℵ‘suc 𝐴)) ≼ (ℵ‘𝐴) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴)))
68	65, 67	syl6 35	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴))))
69		domentr 9073	. . . . . . . . . 10 ⊢ ((((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴)) ∧ ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴)) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
70	69	expcom 413	. . . . . . . . 9 ⊢ (((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴) → (((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴)) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ (ℵ‘𝐴)))
71	61, 68, 70	sylsyld 61	. . . . . . . 8 ⊢ (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ (ℵ‘𝐴)))
72	71	imp 406	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
73		domtr 9067	. . . . . . 7 ⊢ (((ℵ‘suc 𝐴) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ∧ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ (ℵ‘𝐴)) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝐴))
74	56, 72, 73	syl2anc 583	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝐴))
75		domnsym 9165	. . . . . 6 ⊢ ((ℵ‘suc 𝐴) ≼ (ℵ‘𝐴) → ¬ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
76	74, 75	syl 17	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → ¬ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
77	76	ex 412	. . . 4 ⊢ (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)))
78	1, 77	mt2d 136	. . 3 ⊢ (𝐴 ∈ On → ¬ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))
79		cfon 10324	. . . . 5 ⊢ (cf‘(ℵ‘suc 𝐴)) ∈ On
80		cfle 10323	. . . . . 6 ⊢ (cf‘(ℵ‘suc 𝐴)) ⊆ (ℵ‘suc 𝐴)
81		onsseleq 6436	. . . . . 6 ⊢ (((cf‘(ℵ‘suc 𝐴)) ∈ On ∧ (ℵ‘suc 𝐴) ∈ On) → ((cf‘(ℵ‘suc 𝐴)) ⊆ (ℵ‘suc 𝐴) ↔ ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) ∨ (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))))
82	80, 81	mpbii 233	. . . . 5 ⊢ (((cf‘(ℵ‘suc 𝐴)) ∈ On ∧ (ℵ‘suc 𝐴) ∈ On) → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) ∨ (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)))
83	79, 2, 82	mp2an 691	. . . 4 ⊢ ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) ∨ (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
84	83	ori 860	. . 3 ⊢ (¬ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
85	78, 84	syl 17	. 2 ⊢ (𝐴 ∈ On → (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
86		cf0 10320	. . 3 ⊢ (cf‘∅) = ∅
87		alephfnon 10134	. . . . . . . 8 ⊢ ℵ Fn On
88	87	fndmi 6683	. . . . . . 7 ⊢ dom ℵ = On
89	88	eleq2i 2836	. . . . . 6 ⊢ (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
90		onsucb 7853	. . . . . 6 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
91	89, 90	bitr4i 278	. . . . 5 ⊢ (suc 𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
92		ndmfv 6955	. . . . 5 ⊢ (¬ suc 𝐴 ∈ dom ℵ → (ℵ‘suc 𝐴) = ∅)
93	91, 92	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ On → (ℵ‘suc 𝐴) = ∅)
94	93	fveq2d 6924	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘(ℵ‘suc 𝐴)) = (cf‘∅))
95	86, 94, 93	3eqtr4a 2806	. 2 ⊢ (¬ 𝐴 ∈ On → (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
96	85, 95	pm2.61i 182	1 ⊢ (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∨ wo 846   = wceq 1537  ∃wex 1777   ∈ wcel 2108  ∀wral 3067  ∃wrex 3076  Vcvv 3488   ⊆ wss 3976  ∅c0 4352  ∪ ciun 5015   class class class wbr 5166   × cxp 5698  dom cdm 5700  Oncon0 6395  Lim wlim 6396  suc csuc 6397  ⟶wf 6569  –1-1→wf1 6570  ‘cfv 6573  ωcom 7903   ≈ cen 9000   ≼ cdom 9001   ≺ csdm 9002  cardccrd 10004  ℵcale 10005  cfccf 10006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-inf2 9710  ax-ac2 10532

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-se 5653  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-isom 6582  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-1o 8522  df-er 8763  df-map 8886  df-en 9004  df-dom 9005  df-sdom 9006  df-fin 9007  df-oi 9579  df-har 9626  df-card 10008  df-aleph 10009  df-cf 10010  df-acn 10011  df-ac 10185

This theorem is referenced by:  pwcfsdom  10652  minregex  43496