Theorem alephreg 10599

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 10090	. . . 4 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
2		alephon 10086	. . . . . . . . 9 ⊢ (ℵ‘suc 𝐴) ∈ On
3		cff1 10275	. . . . . . . . 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 6904	. . . . . . . . . . . . 13 ⊢ (cf‘(ℵ‘suc 𝐴)) ∈ V
6		fvex 6904	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑦) ∈ V
7	6	sucex 7803	. . . . . . . . . . . . 13 ⊢ suc (𝑓‘𝑦) ∈ V
8	5, 7	iunex 7966	. . . . . . . . . . . 12 ⊢ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) ∈ V
9		f1f 6787	. . . . . . . . . . . . . 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 6477	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (ℵ‘suc 𝐴) → 𝑥 ∈ On)
13		ffvelcdm 7085	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴))
14		onelon 6388	. . . . . . . . . . . . . . . . . . . . . . 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 6453	. . . . . . . . . . . . . . . . . . . . . 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 3171	. . . . . . . . . . . . . . . . . . 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 4995	. . . . . . . . . . . . . . . . . . 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 3170	. . . . . . . . . . . . . . 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 3966	. . . . . . . . . . . . . . 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 9014	. . . . . . . . . . . 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 7808	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
33		alephislim 10100	. . . . . . . . . . . . . . . . . . 19 ⊢ (suc 𝐴 ∈ On ↔ Lim (ℵ‘suc 𝐴))
34		limsuc 7847	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim (ℵ‘suc 𝐴) → ((𝑓‘𝑦) ∈ (ℵ‘suc 𝐴) ↔ suc (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴)))
35	33, 34	sylbi 216	. . . . . . . . . . . . . . . . . 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 5145	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = suc (𝑓‘𝑦) → (𝑧 ≺ (ℵ‘suc 𝐴) ↔ suc (𝑓‘𝑦) ≺ (ℵ‘suc 𝐴)))
38		alephcard 10087	. . . . . . . . . . . . . . . . . . . 20 ⊢ (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
39		iscard 9992	. . . . . . . . . . . . . . . . . . . . 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 3571	. . . . . . . . . . . . . . . . . 18 ⊢ (suc (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓‘𝑦) ≺ (ℵ‘suc 𝐴))
43		alephsucdom 10096	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ On → (suc (𝑓‘𝑦) ≼ (ℵ‘𝐴) ↔ suc (𝑓‘𝑦) ≺ (ℵ‘suc 𝐴)))
44	42, 43	imbitrrid 245	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ On → (suc (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓‘𝑦) ≼ (ℵ‘𝐴)))
45	36, 44	sylbid 239	. . . . . . . . . . . . . . . 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 3140	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → ∀𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) ≼ (ℵ‘𝐴))
49		iundom 10559	. . . . . . . . . . . . 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 9021	. . . . . . . . . . 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 1929	. . . . . . . 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 10099	. . . . . . . . . 10 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
58		alephon 10086	. . . . . . . . . . 11 ⊢ (ℵ‘𝐴) ∈ On
59		infxpen 10031	. . . . . . . . . . 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 216	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
62		breq1 5145	. . . . . . . . . . . 12 ⊢ (𝑧 = (cf‘(ℵ‘suc 𝐴)) → (𝑧 ≺ (ℵ‘suc 𝐴) ↔ (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴)))
63	62, 41	vtoclri 3571	. . . . . . . . . . 11 ⊢ ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴))
64		alephsucdom 10096	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ≼ (ℵ‘𝐴) ↔ (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴)))
65	63, 64	imbitrrid 245	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) ≼ (ℵ‘𝐴)))
66		fvex 6904	. . . . . . . . . . 11 ⊢ (ℵ‘𝐴) ∈ V
67	66	xpdom1 9089	. . . . . . . . . 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 9027	. . . . . . . . . 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 9021	. . . . . . 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 9117	. . . . . 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 10272	. . . . 5 ⊢ (cf‘(ℵ‘suc 𝐴)) ∈ On
80		cfle 10271	. . . . . 6 ⊢ (cf‘(ℵ‘suc 𝐴)) ⊆ (ℵ‘suc 𝐴)
81		onsseleq 6404	. . . . . 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 232	. . . . 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 10268	. . 3 ⊢ (cf‘∅) = ∅
87		alephfnon 10082	. . . . . . . 8 ⊢ ℵ Fn On
88	87	fndmi 6652	. . . . . . 7 ⊢ dom ℵ = On
89	88	eleq2i 2820	. . . . . 6 ⊢ (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
90		onsucb 7814	. . . . . 6 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
91	89, 90	bitr4i 278	. . . . 5 ⊢ (suc 𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
92		ndmfv 6926	. . . . 5 ⊢ (¬ suc 𝐴 ∈ dom ℵ → (ℵ‘suc 𝐴) = ∅)
93	91, 92	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ On → (ℵ‘suc 𝐴) = ∅)
94	93	fveq2d 6895	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘(ℵ‘suc 𝐴)) = (cf‘∅))
95	86, 94, 93	3eqtr4a 2793	. 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 205   ∧ wa 395   ∨ wo 846   = wceq 1534  ∃wex 1774   ∈ wcel 2099  ∀wral 3056  ∃wrex 3065  Vcvv 3469   ⊆ wss 3944  ∅c0 4318  ∪ ciun 4991   class class class wbr 5142   × cxp 5670  dom cdm 5672  Oncon0 6363  Lim wlim 6364  suc csuc 6365  ⟶wf 6538  –1-1→wf1 6539  ‘cfv 6542  ωcom 7864   ≈ cen 8954   ≼ cdom 8955   ≺ csdm 8956  cardccrd 9952  ℵcale 9953  cfccf 9954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734  ax-inf2 9658  ax-ac2 10480

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7865  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-oi 9527  df-har 9574  df-card 9956  df-aleph 9957  df-cf 9958  df-acn 9959  df-ac 10133

This theorem is referenced by:  pwcfsdom  10600  minregex  42936