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Theorem alephreg 9739
 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.
StepHypRef Expression
1 alephordilem1 9229 . . . 4 (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
2 alephon 9225 . . . . . . . . 9 (ℵ‘suc 𝐴) ∈ On
3 cff1 9415 . . . . . . . . 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 ⊆ (𝑓𝑦)))
42, 3ax-mp 5 . . . . . . . 8 𝑓(𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓𝑦))
5 fvex 6459 . . . . . . . . . . . . 13 (cf‘(ℵ‘suc 𝐴)) ∈ V
6 fvex 6459 . . . . . . . . . . . . . 14 (𝑓𝑦) ∈ V
76sucex 7289 . . . . . . . . . . . . 13 suc (𝑓𝑦) ∈ V
85, 7iunex 7425 . . . . . . . . . . . 12 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ∈ V
9 f1f 6351 . . . . . . . . . . . . . 14 (𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) → 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴))
109ad2antrr 716 . . . . . . . . . . . . 13 (((𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓𝑦)) ∧ (𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))) → 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴))
11 simplr 759 . . . . . . . . . . . . 13 (((𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓𝑦)) ∧ (𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))) → ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓𝑦))
122oneli 6083 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℵ‘suc 𝐴) → 𝑥 ∈ On)
13 ffvelrn 6621 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑓𝑦) ∈ (ℵ‘suc 𝐴))
14 onelon 6001 . . . . . . . . . . . . . . . . . . . . . . 23 (((ℵ‘suc 𝐴) ∈ On ∧ (𝑓𝑦) ∈ (ℵ‘suc 𝐴)) → (𝑓𝑦) ∈ On)
152, 13, 14sylancr 581 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑓𝑦) ∈ On)
16 onsssuc 6063 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ On ∧ (𝑓𝑦) ∈ On) → (𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 ∈ suc (𝑓𝑦)))
1715, 16sylan2 586 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ On ∧ (𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴)))) → (𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 ∈ suc (𝑓𝑦)))
1817anassrs 461 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 ∈ suc (𝑓𝑦)))
1918rexbidva 3234 . . . . . . . . . . . . . . . . . . 19 ((𝑥 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → (∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓𝑦) ↔ ∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ∈ suc (𝑓𝑦)))
20 eliun 4757 . . . . . . . . . . . . . . . . . . 19 (𝑥 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ↔ ∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ∈ suc (𝑓𝑦))
2119, 20syl6bbr 281 . . . . . . . . . . . . . . . . . 18 ((𝑥 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → (∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦)))
2221ancoms 452 . . . . . . . . . . . . . . . . 17 ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑥 ∈ On) → (∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦)))
2312, 22sylan2 586 . . . . . . . . . . . . . . . 16 ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑥 ∈ (ℵ‘suc 𝐴)) → (∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦)))
2423ralbidva 3167 . . . . . . . . . . . . . . 15 (𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) → (∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓𝑦) ↔ ∀𝑥 ∈ (ℵ‘suc 𝐴)𝑥 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦)))
25 dfss3 3810 . . . . . . . . . . . . . . 15 ((ℵ‘suc 𝐴) ⊆ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ↔ ∀𝑥 ∈ (ℵ‘suc 𝐴)𝑥 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦))
2624, 25syl6bbr 281 . . . . . . . . . . . . . 14 (𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) → (∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓𝑦) ↔ (ℵ‘suc 𝐴) ⊆ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦)))
2726biimpa 470 . . . . . . . . . . . . 13 ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓𝑦)) → (ℵ‘suc 𝐴) ⊆ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦))
2810, 11, 27syl2anc 579 . . . . . . . . . . . 12 (((𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓𝑦)) ∧ (𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))) → (ℵ‘suc 𝐴) ⊆ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦))
29 ssdomg 8287 . . . . . . . . . . . 12 ( 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ∈ V → ((ℵ‘suc 𝐴) ⊆ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) → (ℵ‘suc 𝐴) ≼ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦)))
308, 28, 29mpsyl 68 . . . . . . . . . . 11 (((𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓𝑦)) ∧ (𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))) → (ℵ‘suc 𝐴) ≼ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦))
31 simprl 761 . . . . . . . . . . . 12 (((𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓𝑦)) ∧ (𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))) → 𝐴 ∈ On)
32 suceloni 7291 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ On → suc 𝐴 ∈ On)
33 alephislim 9239 . . . . . . . . . . . . . . . . . . 19 (suc 𝐴 ∈ On ↔ Lim (ℵ‘suc 𝐴))
34 limsuc 7327 . . . . . . . . . . . . . . . . . . 19 (Lim (ℵ‘suc 𝐴) → ((𝑓𝑦) ∈ (ℵ‘suc 𝐴) ↔ suc (𝑓𝑦) ∈ (ℵ‘suc 𝐴)))
3533, 34sylbi 209 . . . . . . . . . . . . . . . . . 18 (suc 𝐴 ∈ On → ((𝑓𝑦) ∈ (ℵ‘suc 𝐴) ↔ suc (𝑓𝑦) ∈ (ℵ‘suc 𝐴)))
3632, 35syl 17 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ On → ((𝑓𝑦) ∈ (ℵ‘suc 𝐴) ↔ suc (𝑓𝑦) ∈ (ℵ‘suc 𝐴)))
37 breq1 4889 . . . . . . . . . . . . . . . . . . 19 (𝑧 = suc (𝑓𝑦) → (𝑧 ≺ (ℵ‘suc 𝐴) ↔ suc (𝑓𝑦) ≺ (ℵ‘suc 𝐴)))
38 alephcard 9226 . . . . . . . . . . . . . . . . . . . 20 (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
39 iscard 9134 . . . . . . . . . . . . . . . . . . . . 21 ((card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴) ↔ ((ℵ‘suc 𝐴) ∈ On ∧ ∀𝑧 ∈ (ℵ‘suc 𝐴)𝑧 ≺ (ℵ‘suc 𝐴)))
4039simprbi 492 . . . . . . . . . . . . . . . . . . . 20 ((card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴) → ∀𝑧 ∈ (ℵ‘suc 𝐴)𝑧 ≺ (ℵ‘suc 𝐴))
4138, 40ax-mp 5 . . . . . . . . . . . . . . . . . . 19 𝑧 ∈ (ℵ‘suc 𝐴)𝑧 ≺ (ℵ‘suc 𝐴)
4237, 41vtoclri 3485 . . . . . . . . . . . . . . . . . 18 (suc (𝑓𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓𝑦) ≺ (ℵ‘suc 𝐴))
43 alephsucdom 9235 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ On → (suc (𝑓𝑦) ≼ (ℵ‘𝐴) ↔ suc (𝑓𝑦) ≺ (ℵ‘suc 𝐴)))
4442, 43syl5ibr 238 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ On → (suc (𝑓𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓𝑦) ≼ (ℵ‘𝐴)))
4536, 44sylbid 232 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On → ((𝑓𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓𝑦) ≼ (ℵ‘𝐴)))
4613, 45syl5 34 . . . . . . . . . . . . . . 15 (𝐴 ∈ On → ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → suc (𝑓𝑦) ≼ (ℵ‘𝐴)))
4746expdimp 446 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → (𝑦 ∈ (cf‘(ℵ‘suc 𝐴)) → suc (𝑓𝑦) ≼ (ℵ‘𝐴)))
4847ralrimiv 3147 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → ∀𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ≼ (ℵ‘𝐴))
49 iundom 9699 . . . . . . . . . . . . 13 (((cf‘(ℵ‘suc 𝐴)) ∈ V ∧ ∀𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ≼ (ℵ‘𝐴)) → 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)))
505, 48, 49sylancr 581 . . . . . . . . . . . 12 ((𝐴 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)))
5131, 10, 50syl2anc 579 . . . . . . . . . . 11 (((𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓𝑦)) ∧ (𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))) → 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)))
52 domtr 8294 . . . . . . . . . . 11 (((ℵ‘suc 𝐴) ≼ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴))) → (ℵ‘suc 𝐴) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)))
5330, 51, 52syl2anc 579 . . . . . . . . . 10 (((𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) ∧ ∀𝑥 ∈ (ℵ‘suc 𝐴)∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ⊆ (𝑓𝑦)) ∧ (𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))) → (ℵ‘suc 𝐴) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)))
5453expcom 404 . . . . . . . . 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 ⊆ (𝑓𝑦)) → (ℵ‘suc 𝐴) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴))))
5554exlimdv 1976 . . . . . . . 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 ⊆ (𝑓𝑦)) → (ℵ‘suc 𝐴) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴))))
564, 55mpi 20 . . . . . . 7 ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → (ℵ‘suc 𝐴) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)))
57 alephgeom 9238 . . . . . . . . . 10 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
58 alephon 9225 . . . . . . . . . . 11 (ℵ‘𝐴) ∈ On
59 infxpen 9170 . . . . . . . . . . 11 (((ℵ‘𝐴) ∈ On ∧ ω ⊆ (ℵ‘𝐴)) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
6058, 59mpan 680 . . . . . . . . . 10 (ω ⊆ (ℵ‘𝐴) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
6157, 60sylbi 209 . . . . . . . . 9 (𝐴 ∈ On → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
62 breq1 4889 . . . . . . . . . . . 12 (𝑧 = (cf‘(ℵ‘suc 𝐴)) → (𝑧 ≺ (ℵ‘suc 𝐴) ↔ (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴)))
6362, 41vtoclri 3485 . . . . . . . . . . 11 ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴))
64 alephsucdom 9235 . . . . . . . . . . 11 (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ≼ (ℵ‘𝐴) ↔ (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴)))
6563, 64syl5ibr 238 . . . . . . . . . 10 (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) ≼ (ℵ‘𝐴)))
66 fvex 6459 . . . . . . . . . . 11 (ℵ‘𝐴) ∈ V
6766xpdom1 8347 . . . . . . . . . 10 ((cf‘(ℵ‘suc 𝐴)) ≼ (ℵ‘𝐴) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴)))
6865, 67syl6 35 . . . . . . . . 9 (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴))))
69 domentr 8300 . . . . . . . . . 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))
7069expcom 404 . . . . . . . . 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)))
7161, 68, 70sylsyld 61 . . . . . . . 8 (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ (ℵ‘𝐴)))
7271imp 397 . . . . . . 7 ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
73 domtr 8294 . . . . . . 7 (((ℵ‘suc 𝐴) ≼ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ∧ ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ (ℵ‘𝐴)) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝐴))
7456, 72, 73syl2anc 579 . . . . . 6 ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝐴))
75 domnsym 8374 . . . . . 6 ((ℵ‘suc 𝐴) ≼ (ℵ‘𝐴) → ¬ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
7674, 75syl 17 . . . . 5 ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → ¬ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
7776ex 403 . . . 4 (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)))
781, 77mt2d 134 . . 3 (𝐴 ∈ On → ¬ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))
79 cfon 9412 . . . . 5 (cf‘(ℵ‘suc 𝐴)) ∈ On
80 cfle 9411 . . . . . 6 (cf‘(ℵ‘suc 𝐴)) ⊆ (ℵ‘suc 𝐴)
81 onsseleq 6017 . . . . . 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))))
8280, 81mpbii 225 . . . . 5 (((cf‘(ℵ‘suc 𝐴)) ∈ On ∧ (ℵ‘suc 𝐴) ∈ On) → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) ∨ (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)))
8379, 2, 82mp2an 682 . . . 4 ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) ∨ (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
8483ori 850 . . 3 (¬ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
8578, 84syl 17 . 2 (𝐴 ∈ On → (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
86 cf0 9408 . . 3 (cf‘∅) = ∅
87 alephfnon 9221 . . . . . . . 8 ℵ Fn On
88 fndm 6235 . . . . . . . 8 (ℵ Fn On → dom ℵ = On)
8987, 88ax-mp 5 . . . . . . 7 dom ℵ = On
9089eleq2i 2851 . . . . . 6 (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
91 sucelon 7295 . . . . . 6 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
9290, 91bitr4i 270 . . . . 5 (suc 𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
93 ndmfv 6476 . . . . 5 (¬ suc 𝐴 ∈ dom ℵ → (ℵ‘suc 𝐴) = ∅)
9492, 93sylnbir 323 . . . 4 𝐴 ∈ On → (ℵ‘suc 𝐴) = ∅)
9594fveq2d 6450 . . 3 𝐴 ∈ On → (cf‘(ℵ‘suc 𝐴)) = (cf‘∅))
9686, 95, 943eqtr4a 2840 . 2 𝐴 ∈ On → (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
9785, 96pm2.61i 177 1 (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 198   ∧ wa 386   ∨ wo 836   = wceq 1601  ∃wex 1823   ∈ wcel 2107  ∀wral 3090  ∃wrex 3091  Vcvv 3398   ⊆ wss 3792  ∅c0 4141  ∪ ciun 4753   class class class wbr 4886   × cxp 5353  dom cdm 5355  Oncon0 5976  Lim wlim 5977  suc csuc 5978   Fn wfn 6130  ⟶wf 6131  –1-1→wf1 6132  ‘cfv 6135  ωcom 7343   ≈ cen 8238   ≼ cdom 8239   ≺ csdm 8240  cardccrd 9094  ℵcale 9095  cfccf 9096 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-ac2 9620 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-int 4711  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-se 5315  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-isom 6144  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-oi 8704  df-har 8752  df-card 9098  df-aleph 9099  df-cf 9100  df-acn 9101  df-ac 9272 This theorem is referenced by:  pwcfsdom  9740
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