Theorem alephreg 10622

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 10113	. . . 4 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
2		alephon 10109	. . . . . . . . 9 ⊢ (ℵ‘suc 𝐴) ∈ On
3		cff1 10298	. . . . . . . . 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 6919	. . . . . . . . . . . . 13 ⊢ (cf‘(ℵ‘suc 𝐴)) ∈ V
6		fvex 6919	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑦) ∈ V
7	6	sucex 7826	. . . . . . . . . . . . 13 ⊢ suc (𝑓‘𝑦) ∈ V
8	5, 7	iunex 7993	. . . . . . . . . . . 12 ⊢ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) ∈ V
9		f1f 6804	. . . . . . . . . . . . . 14 ⊢ (𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) → 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴))
10	9	ad2antrr 726	. . . . . . . . . . . . 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 769	. . . . . . . . . . . . 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 6498	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (ℵ‘suc 𝐴) → 𝑥 ∈ On)
13		ffvelcdm 7101	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴))
14		onelon 6409	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((ℵ‘suc 𝐴) ∈ On ∧ (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴)) → (𝑓‘𝑦) ∈ On)
15	2, 13, 14	sylancr 587	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑓‘𝑦) ∈ On)
16		onsssuc 6474	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ On ∧ (𝑓‘𝑦) ∈ On) → (𝑥 ⊆ (𝑓‘𝑦) ↔ 𝑥 ∈ suc (𝑓‘𝑦)))
17	15, 16	sylan2 593	. . . . . . . . . . . . . . . . . . . . 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 3177	. . . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . 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 3176	. . . . . . . . . . . . . . 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 3972	. . . . . . . . . . . . . . 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 584	. . . . . . . . . . . 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 9040	. . . . . . . . . . . 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 771	. . . . . . . . . . . 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 7831	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
33		alephislim 10123	. . . . . . . . . . . . . . . . . . 19 ⊢ (suc 𝐴 ∈ On ↔ Lim (ℵ‘suc 𝐴))
34		limsuc 7870	. . . . . . . . . . . . . . . . . . 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 5146	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = suc (𝑓‘𝑦) → (𝑧 ≺ (ℵ‘suc 𝐴) ↔ suc (𝑓‘𝑦) ≺ (ℵ‘suc 𝐴)))
38		alephcard 10110	. . . . . . . . . . . . . . . . . . . 20 ⊢ (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
39		iscard 10015	. . . . . . . . . . . . . . . . . . . . 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 3590	. . . . . . . . . . . . . . . . . 18 ⊢ (suc (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓‘𝑦) ≺ (ℵ‘suc 𝐴))
43		alephsucdom 10119	. . . . . . . . . . . . . . . . . 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 3145	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → ∀𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) ≼ (ℵ‘𝐴))
49		iundom 10582	. . . . . . . . . . . . 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 587	. . . . . . . . . . . 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 584	. . . . . . . . . . 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 9047	. . . . . . . . . . 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 584	. . . . . . . . . 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 1933	. . . . . . . 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 10122	. . . . . . . . . 10 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
58		alephon 10109	. . . . . . . . . . 11 ⊢ (ℵ‘𝐴) ∈ On
59		infxpen 10054	. . . . . . . . . . 11 ⊢ (((ℵ‘𝐴) ∈ On ∧ ω ⊆ (ℵ‘𝐴)) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
60	58, 59	mpan 690	. . . . . . . . . 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 5146	. . . . . . . . . . . 12 ⊢ (𝑧 = (cf‘(ℵ‘suc 𝐴)) → (𝑧 ≺ (ℵ‘suc 𝐴) ↔ (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴)))
63	62, 41	vtoclri 3590	. . . . . . . . . . 11 ⊢ ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴))
64		alephsucdom 10119	. . . . . . . . . . 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 6919	. . . . . . . . . . 11 ⊢ (ℵ‘𝐴) ∈ V
67	66	xpdom1 9111	. . . . . . . . . 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 9053	. . . . . . . . . 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 9047	. . . . . . 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 584	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝐴))
75		domnsym 9139	. . . . . 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 10295	. . . . 5 ⊢ (cf‘(ℵ‘suc 𝐴)) ∈ On
80		cfle 10294	. . . . . 6 ⊢ (cf‘(ℵ‘suc 𝐴)) ⊆ (ℵ‘suc 𝐴)
81		onsseleq 6425	. . . . . 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 692	. . . 4 ⊢ ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) ∨ (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
84	83	ori 862	. . 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 10291	. . 3 ⊢ (cf‘∅) = ∅
87		alephfnon 10105	. . . . . . . 8 ⊢ ℵ Fn On
88	87	fndmi 6672	. . . . . . 7 ⊢ dom ℵ = On
89	88	eleq2i 2833	. . . . . 6 ⊢ (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
90		onsucb 7837	. . . . . 6 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
91	89, 90	bitr4i 278	. . . . 5 ⊢ (suc 𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
92		ndmfv 6941	. . . . 5 ⊢ (¬ suc 𝐴 ∈ dom ℵ → (ℵ‘suc 𝐴) = ∅)
93	91, 92	sylnbir 331	. . . 4 ⊢ (¬ 𝐴 ∈ On → (ℵ‘suc 𝐴) = ∅)
94	93	fveq2d 6910	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘(ℵ‘suc 𝐴)) = (cf‘∅))
95	86, 94, 93	3eqtr4a 2803	. 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 848   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ⊆ wss 3951  ∅c0 4333  ∪ ciun 4991   class class class wbr 5143   × cxp 5683  dom cdm 5685  Oncon0 6384  Lim wlim 6385  suc csuc 6386  ⟶wf 6557  –1-1→wf1 6558  ‘cfv 6561  ωcom 7887   ≈ cen 8982   ≼ cdom 8983   ≺ csdm 8984  cardccrd 9975  ℵcale 9976  cfccf 9977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-ac2 10503

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-oi 9550  df-har 9597  df-card 9979  df-aleph 9980  df-cf 9981  df-acn 9982  df-ac 10156

This theorem is referenced by:  pwcfsdom  10623  minregex  43547