Theorem alephreg 10668

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 10153	. . . 4 ⊢ (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
2		alephon 10149	. . . . . . . . 9 ⊢ (ℵ‘suc 𝐴) ∈ On
3		cff1 10337	. . . . . . . . 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 6898	. . . . . . . . . . . . 13 ⊢ (cf‘(ℵ‘suc 𝐴)) ∈ V
6		fvex 6898	. . . . . . . . . . . . . 14 ⊢ (𝑓‘𝑦) ∈ V
7	6	sucex 7820	. . . . . . . . . . . . 13 ⊢ suc (𝑓‘𝑦) ∈ V
8	5, 7	iunex 7980	. . . . . . . . . . . 12 ⊢ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) ∈ V
9		f1f 6778	. . . . . . . . . . . . . 14 ⊢ (𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) → 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴))
10	9	ad2antrr 739	. . . . . . . . . . . . 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 781	. . . . . . . . . . . . 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 6478	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (ℵ‘suc 𝐴) → 𝑥 ∈ On)
13		ffvelcdm 7081	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴))
14		onelon 6387	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((ℵ‘suc 𝐴) ∈ On ∧ (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴)) → (𝑓‘𝑦) ∈ On)
15	2, 13, 14	sylancr 599	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑓‘𝑦) ∈ On)
16		onsssuc 6455	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ On ∧ (𝑓‘𝑦) ∈ On) → (𝑥 ⊆ (𝑓‘𝑦) ↔ 𝑥 ∈ suc (𝑓‘𝑦)))
17	15, 16	sylan2 605	. . . . . . . . . . . . . . . . . . . . 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 473	. . . . . . . . . . . . . . . . . . . 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 3184	. . . . . . . . . . . . . . . . . . 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 4954	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) ↔ ∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ∈ suc (𝑓‘𝑦))
21	19, 20	bitr4di 292	. . . . . . . . . . . . . . . . . 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 464	. . . . . . . . . . . . . . . . 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 605	. . . . . . . . . . . . . . . 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 3183	. . . . . . . . . . . . . . 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 3897	. . . . . . . . . . . . . . 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 292	. . . . . . . . . . . . . 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 482	. . . . . . . . . . . . 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 596	. . . . . . . . . . . 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 9028	. . . . . . . . . . . 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 69	. . . . . . . . . . 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 783	. . . . . . . . . . . 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 7824	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ On → suc 𝐴 ∈ On)
33		alephislim 10163	. . . . . . . . . . . . . . . . . . 19 ⊢ (suc 𝐴 ∈ On ↔ Lim (ℵ‘suc 𝐴))
34		limsuc 7860	. . . . . . . . . . . . . . . . . . 19 ⊢ (Lim (ℵ‘suc 𝐴) → ((𝑓‘𝑦) ∈ (ℵ‘suc 𝐴) ↔ suc (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴)))
35	33, 34	sylbi 220	. . . . . . . . . . . . . . . . . 18 ⊢ (suc 𝐴 ∈ On → ((𝑓‘𝑦) ∈ (ℵ‘suc 𝐴) ↔ suc (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴)))
36	32, 35	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ On → ((𝑓‘𝑦) ∈ (ℵ‘suc 𝐴) ↔ suc (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴)))
37		breq1 5105	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = suc (𝑓‘𝑦) → (𝑧 ≺ (ℵ‘suc 𝐴) ↔ suc (𝑓‘𝑦) ≺ (ℵ‘suc 𝐴)))
38		alephcard 10150	. . . . . . . . . . . . . . . . . . . 20 ⊢ (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
39		iscard 10057	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴) ↔ ((ℵ‘suc 𝐴) ∈ On ∧ ∀𝑧 ∈ (ℵ‘suc 𝐴)𝑧 ≺ (ℵ‘suc 𝐴)))
40	39	simprbi 503	. . . . . . . . . . . . . . . . . . . 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 3544	. . . . . . . . . . . . . . . . . 18 ⊢ (suc (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓‘𝑦) ≺ (ℵ‘suc 𝐴))
43		alephsucdom 10159	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∈ On → (suc (𝑓‘𝑦) ≼ (ℵ‘𝐴) ↔ suc (𝑓‘𝑦) ≺ (ℵ‘suc 𝐴)))
44	42, 43	imbitrrid 249	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴 ∈ On → (suc (𝑓‘𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓‘𝑦) ≼ (ℵ‘𝐴)))
45	36, 44	sylbid 243	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ∈ On → ((𝑓‘𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓‘𝑦) ≼ (ℵ‘𝐴)))
46	13, 45	syl5 35	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∈ On → ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → suc (𝑓‘𝑦) ≼ (ℵ‘𝐴)))
47	46	expdimp 458	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → (𝑦 ∈ (cf‘(ℵ‘suc 𝐴)) → suc (𝑓‘𝑦) ≼ (ℵ‘𝐴)))
48	47	ralrimiv 3153	. . . . . . . . . . . . 13 ⊢ ((𝐴 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → ∀𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓‘𝑦) ≼ (ℵ‘𝐴))
49		iundom 10627	. . . . . . . . . . . . 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 599	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 9035	. . . . . . . . . . 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 596	. . . . . . . . . 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 419	. . . . . . . . 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 1965	. . . . . . . 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 21	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → (ℵ‘suc 𝐴) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)))
57		alephgeom 10162	. . . . . . . . . 10 ⊢ (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
58		alephon 10149	. . . . . . . . . . 11 ⊢ (ℵ‘𝐴) ∈ On
59		infxpen 10094	. . . . . . . . . . 11 ⊢ (((ℵ‘𝐴) ∈ On ∧ ω ⊆ (ℵ‘𝐴)) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
60	58, 59	mpan 703	. . . . . . . . . 10 ⊢ (ω ⊆ (ℵ‘𝐴) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
61	57, 60	sylbi 220	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
62		breq1 5105	. . . . . . . . . . . 12 ⊢ (𝑧 = (cf‘(ℵ‘suc 𝐴)) → (𝑧 ≺ (ℵ‘suc 𝐴) ↔ (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴)))
63	62, 41	vtoclri 3544	. . . . . . . . . . 11 ⊢ ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴))
64		alephsucdom 10159	. . . . . . . . . . 11 ⊢ (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ≼ (ℵ‘𝐴) ↔ (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴)))
65	63, 64	imbitrrid 249	. . . . . . . . . 10 ⊢ (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) ≼ (ℵ‘𝐴)))
66		fvex 6898	. . . . . . . . . . 11 ⊢ (ℵ‘𝐴) ∈ V
67	66	xpdom1 9096	. . . . . . . . . 10 ⊢ ((cf‘(ℵ‘suc 𝐴)) ≼ (ℵ‘𝐴) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴)))
68	65, 67	syl6 36	. . . . . . . . 9 ⊢ (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴))))
69		domentr 9041	. . . . . . . . . 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 419	. . . . . . . . 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 62	. . . . . . . 8 ⊢ (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ (ℵ‘𝐴)))
72	71	imp 412	. . . . . . 7 ⊢ ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
73		domtr 9035	. . . . . . 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 596	. . . . . 6 ⊢ ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝐴))
75		domnsym 9123	. . . . . 6 ⊢ ((ℵ‘suc 𝐴) ≼ (ℵ‘𝐴) → ¬ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
76	74, 75	syl 18	. . . . 5 ⊢ ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → ¬ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
77	76	ex 418	. . . 4 ⊢ (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)))
78	1, 77	mt2d 137	. . 3 ⊢ (𝐴 ∈ On → ¬ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))
79		cfon 10333	. . . . 5 ⊢ (cf‘(ℵ‘suc 𝐴)) ∈ On
80		cfle 10332	. . . . . 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 236	. . . . 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 705	. . . 4 ⊢ ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) ∨ (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
84	83	ori 875	. . 3 ⊢ (¬ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
85	78, 84	syl 18	. 2 ⊢ (𝐴 ∈ On → (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
86		cf0 10329	. . 3 ⊢ (cf‘∅) = ∅
87		alephfnon 10145	. . . . . . . 8 ⊢ ℵ Fn On
88	87	fndmi 6643	. . . . . . 7 ⊢ dom ℵ = On
89	88	eleq2i 2852	. . . . . 6 ⊢ (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
90		onsucb 7828	. . . . . 6 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
91	89, 90	bitr4i 281	. . . . 5 ⊢ (suc 𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
92		ndmfv 6917	. . . . 5 ⊢ (¬ suc 𝐴 ∈ dom ℵ → (ℵ‘suc 𝐴) = ∅)
93	91, 92	sylnbir 334	. . . 4 ⊢ (¬ 𝐴 ∈ On → (ℵ‘suc 𝐴) = ∅)
94	93	fveq2d 6889	. . 3 ⊢ (¬ 𝐴 ∈ On → (cf‘(ℵ‘suc 𝐴)) = (cf‘∅))
95	86, 94, 93	3eqtr4a 2821	. 2 ⊢ (¬ 𝐴 ∈ On → (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
96	85, 95	pm2.61i 184	1 ⊢ (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1569  ∃wex 1811   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ⊆ wss 3895  ∅c0 4278  ∪ ciun 4950   class class class wbr 5102   × cxp 5648  dom cdm 5650  Oncon0 6362  Lim wlim 6363  suc csuc 6364  ⟶wf 6534  –1-1→wf1 6535  ‘cfv 6538  ωcom 7877   ≈ cen 8971   ≼ cdom 8972   ≺ csdm 8973  cardccrd 10017  ℵcale 10018  cfccf 10019

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-inf2 9643  ax-ac2 10542

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8300  df-wrecs 8331  df-recs 8380  df-rdg 8419  df-1o 8477  df-er 8718  df-map 8850  df-en 8975  df-dom 8976  df-sdom 8977  df-fin 8978  df-oi 9505  df-har 9552  df-card 10021  df-aleph 10022  df-cf 10023  df-acn 10024  df-ac 10196

This theorem is used by:  pwcfsdom  10669  minregex  44611