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Theorem alephreg 9992
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 9487 . . . 4 (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
2 alephon 9483 . . . . . . . . 9 (ℵ‘suc 𝐴) ∈ On
3 cff1 9668 . . . . . . . . 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 6676 . . . . . . . . . . . . 13 (cf‘(ℵ‘suc 𝐴)) ∈ V
6 fvex 6676 . . . . . . . . . . . . . 14 (𝑓𝑦) ∈ V
76sucex 7515 . . . . . . . . . . . . 13 suc (𝑓𝑦) ∈ V
85, 7iunex 7658 . . . . . . . . . . . 12 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ∈ V
9 f1f 6568 . . . . . . . . . . . . . 14 (𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) → 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴))
109ad2antrr 722 . . . . . . . . . . . . 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 765 . . . . . . . . . . . . 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 6291 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℵ‘suc 𝐴) → 𝑥 ∈ On)
13 ffvelrn 6841 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑓𝑦) ∈ (ℵ‘suc 𝐴))
14 onelon 6209 . . . . . . . . . . . . . . . . . . . . . . 23 (((ℵ‘suc 𝐴) ∈ On ∧ (𝑓𝑦) ∈ (ℵ‘suc 𝐴)) → (𝑓𝑦) ∈ On)
152, 13, 14sylancr 587 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑓𝑦) ∈ On)
16 onsssuc 6271 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ On ∧ (𝑓𝑦) ∈ On) → (𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 ∈ suc (𝑓𝑦)))
1715, 16sylan2 592 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ On ∧ (𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴)))) → (𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 ∈ suc (𝑓𝑦)))
1817anassrs 468 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 ∈ suc (𝑓𝑦)))
1918rexbidva 3293 . . . . . . . . . . . . . . . . . . 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 4914 . . . . . . . . . . . . . . . . . . 19 (𝑥 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ↔ ∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ∈ suc (𝑓𝑦))
2119, 20syl6bbr 290 . . . . . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . . . . . 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 592 . . . . . . . . . . . . . . . 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 3193 . . . . . . . . . . . . . . 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 3953 . . . . . . . . . . . . . . 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 290 . . . . . . . . . . . . . 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 477 . . . . . . . . . . . . 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 584 . . . . . . . . . . . 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 8543 . . . . . . . . . . . 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 767 . . . . . . . . . . . 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 7517 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ On → suc 𝐴 ∈ On)
33 alephislim 9497 . . . . . . . . . . . . . . . . . . 19 (suc 𝐴 ∈ On ↔ Lim (ℵ‘suc 𝐴))
34 limsuc 7553 . . . . . . . . . . . . . . . . . . 19 (Lim (ℵ‘suc 𝐴) → ((𝑓𝑦) ∈ (ℵ‘suc 𝐴) ↔ suc (𝑓𝑦) ∈ (ℵ‘suc 𝐴)))
3533, 34sylbi 218 . . . . . . . . . . . . . . . . . 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 5060 . . . . . . . . . . . . . . . . . . 19 (𝑧 = suc (𝑓𝑦) → (𝑧 ≺ (ℵ‘suc 𝐴) ↔ suc (𝑓𝑦) ≺ (ℵ‘suc 𝐴)))
38 alephcard 9484 . . . . . . . . . . . . . . . . . . . 20 (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
39 iscard 9392 . . . . . . . . . . . . . . . . . . . . 21 ((card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴) ↔ ((ℵ‘suc 𝐴) ∈ On ∧ ∀𝑧 ∈ (ℵ‘suc 𝐴)𝑧 ≺ (ℵ‘suc 𝐴)))
4039simprbi 497 . . . . . . . . . . . . . . . . . . . 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 3582 . . . . . . . . . . . . . . . . . 18 (suc (𝑓𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓𝑦) ≺ (ℵ‘suc 𝐴))
43 alephsucdom 9493 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ On → (suc (𝑓𝑦) ≼ (ℵ‘𝐴) ↔ suc (𝑓𝑦) ≺ (ℵ‘suc 𝐴)))
4442, 43syl5ibr 247 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ On → (suc (𝑓𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓𝑦) ≼ (ℵ‘𝐴)))
4536, 44sylbid 241 . . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → (𝑦 ∈ (cf‘(ℵ‘suc 𝐴)) → suc (𝑓𝑦) ≼ (ℵ‘𝐴)))
4847ralrimiv 3178 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → ∀𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ≼ (ℵ‘𝐴))
49 iundom 9952 . . . . . . . . . . . . 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 587 . . . . . . . . . . . 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 584 . . . . . . . . . . 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 8550 . . . . . . . . . . 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 584 . . . . . . . . . 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 414 . . . . . . . . 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 1925 . . . . . . . 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 9496 . . . . . . . . . 10 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
58 alephon 9483 . . . . . . . . . . 11 (ℵ‘𝐴) ∈ On
59 infxpen 9428 . . . . . . . . . . 11 (((ℵ‘𝐴) ∈ On ∧ ω ⊆ (ℵ‘𝐴)) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
6058, 59mpan 686 . . . . . . . . . 10 (ω ⊆ (ℵ‘𝐴) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
6157, 60sylbi 218 . . . . . . . . 9 (𝐴 ∈ On → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
62 breq1 5060 . . . . . . . . . . . 12 (𝑧 = (cf‘(ℵ‘suc 𝐴)) → (𝑧 ≺ (ℵ‘suc 𝐴) ↔ (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴)))
6362, 41vtoclri 3582 . . . . . . . . . . 11 ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴))
64 alephsucdom 9493 . . . . . . . . . . 11 (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ≼ (ℵ‘𝐴) ↔ (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴)))
6563, 64syl5ibr 247 . . . . . . . . . 10 (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) ≼ (ℵ‘𝐴)))
66 fvex 6676 . . . . . . . . . . 11 (ℵ‘𝐴) ∈ V
6766xpdom1 8604 . . . . . . . . . 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 8556 . . . . . . . . . 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 414 . . . . . . . . 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 407 . . . . . . 7 ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
73 domtr 8550 . . . . . . 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 584 . . . . . 6 ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝐴))
75 domnsym 8631 . . . . . 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 413 . . . 4 (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)))
781, 77mt2d 138 . . 3 (𝐴 ∈ On → ¬ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))
79 cfon 9665 . . . . 5 (cf‘(ℵ‘suc 𝐴)) ∈ On
80 cfle 9664 . . . . . 6 (cf‘(ℵ‘suc 𝐴)) ⊆ (ℵ‘suc 𝐴)
81 onsseleq 6225 . . . . . 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 234 . . . . 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 688 . . . 4 ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) ∨ (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
8483ori 855 . . 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 9661 . . 3 (cf‘∅) = ∅
87 alephfnon 9479 . . . . . . . 8 ℵ Fn On
88 fndm 6448 . . . . . . . 8 (ℵ Fn On → dom ℵ = On)
8987, 88ax-mp 5 . . . . . . 7 dom ℵ = On
9089eleq2i 2901 . . . . . 6 (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
91 sucelon 7521 . . . . . 6 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
9290, 91bitr4i 279 . . . . 5 (suc 𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
93 ndmfv 6693 . . . . 5 (¬ suc 𝐴 ∈ dom ℵ → (ℵ‘suc 𝐴) = ∅)
9492, 93sylnbir 332 . . . 4 𝐴 ∈ On → (ℵ‘suc 𝐴) = ∅)
9594fveq2d 6667 . . 3 𝐴 ∈ On → (cf‘(ℵ‘suc 𝐴)) = (cf‘∅))
9686, 95, 943eqtr4a 2879 . 2 𝐴 ∈ On → (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
9785, 96pm2.61i 183 1 (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396  wo 841   = wceq 1528  wex 1771  wcel 2105  wral 3135  wrex 3136  Vcvv 3492  wss 3933  c0 4288   ciun 4910   class class class wbr 5057   × cxp 5546  dom cdm 5548  Oncon0 6184  Lim wlim 6185  suc csuc 6186   Fn wfn 6343  wf 6344  1-1wf1 6345  cfv 6348  ωcom 7569  cen 8494  cdom 8495  csdm 8496  cardccrd 9352  cale 9353  cfccf 9354
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-ac2 9873
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-se 5508  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-isom 6357  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-oi 8962  df-har 9010  df-card 9356  df-aleph 9357  df-cf 9358  df-acn 9359  df-ac 9530
This theorem is referenced by:  pwcfsdom  9993
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