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Theorem alephreg 9389
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 8881 . . . 4 (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
2 alephon 8877 . . . . . . . . 9 (ℵ‘suc 𝐴) ∈ On
3 cff1 9065 . . . . . . . . 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 6188 . . . . . . . . . . . . 13 (cf‘(ℵ‘suc 𝐴)) ∈ V
6 fvex 6188 . . . . . . . . . . . . . 14 (𝑓𝑦) ∈ V
76sucex 6996 . . . . . . . . . . . . 13 suc (𝑓𝑦) ∈ V
85, 7iunex 7132 . . . . . . . . . . . 12 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ∈ V
9 f1f 6088 . . . . . . . . . . . . . 14 (𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) → 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴))
109ad2antrr 761 . . . . . . . . . . . . 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 791 . . . . . . . . . . . . 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 5823 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℵ‘suc 𝐴) → 𝑥 ∈ On)
13 ffvelrn 6343 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑓𝑦) ∈ (ℵ‘suc 𝐴))
14 onelon 5736 . . . . . . . . . . . . . . . . . . . . . . 23 (((ℵ‘suc 𝐴) ∈ On ∧ (𝑓𝑦) ∈ (ℵ‘suc 𝐴)) → (𝑓𝑦) ∈ On)
152, 13, 14sylancr 694 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑓𝑦) ∈ On)
16 onsssuc 5801 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ On ∧ (𝑓𝑦) ∈ On) → (𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 ∈ suc (𝑓𝑦)))
1715, 16sylan2 491 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ On ∧ (𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴)))) → (𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 ∈ suc (𝑓𝑦)))
1817anassrs 679 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 ∈ suc (𝑓𝑦)))
1918rexbidva 3045 . . . . . . . . . . . . . . . . . . 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 4515 . . . . . . . . . . . . . . . . . . 19 (𝑥 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ↔ ∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ∈ suc (𝑓𝑦))
2119, 20syl6bbr 278 . . . . . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . . . . . 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 491 . . . . . . . . . . . . . . . 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 2982 . . . . . . . . . . . . . . 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 3585 . . . . . . . . . . . . . . 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 278 . . . . . . . . . . . . . 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 501 . . . . . . . . . . . . 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 692 . . . . . . . . . . . 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 7986 . . . . . . . . . . . 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 793 . . . . . . . . . . . 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 6998 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ On → suc 𝐴 ∈ On)
33 alephislim 8891 . . . . . . . . . . . . . . . . . . 19 (suc 𝐴 ∈ On ↔ Lim (ℵ‘suc 𝐴))
34 limsuc 7034 . . . . . . . . . . . . . . . . . . 19 (Lim (ℵ‘suc 𝐴) → ((𝑓𝑦) ∈ (ℵ‘suc 𝐴) ↔ suc (𝑓𝑦) ∈ (ℵ‘suc 𝐴)))
3533, 34sylbi 207 . . . . . . . . . . . . . . . . . 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 4647 . . . . . . . . . . . . . . . . . . 19 (𝑧 = suc (𝑓𝑦) → (𝑧 ≺ (ℵ‘suc 𝐴) ↔ suc (𝑓𝑦) ≺ (ℵ‘suc 𝐴)))
38 alephcard 8878 . . . . . . . . . . . . . . . . . . . 20 (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
39 iscard 8786 . . . . . . . . . . . . . . . . . . . . 21 ((card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴) ↔ ((ℵ‘suc 𝐴) ∈ On ∧ ∀𝑧 ∈ (ℵ‘suc 𝐴)𝑧 ≺ (ℵ‘suc 𝐴)))
4039simprbi 480 . . . . . . . . . . . . . . . . . . . 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 3278 . . . . . . . . . . . . . . . . . 18 (suc (𝑓𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓𝑦) ≺ (ℵ‘suc 𝐴))
43 alephsucdom 8887 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ On → (suc (𝑓𝑦) ≼ (ℵ‘𝐴) ↔ suc (𝑓𝑦) ≺ (ℵ‘suc 𝐴)))
4442, 43syl5ibr 236 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ On → (suc (𝑓𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓𝑦) ≼ (ℵ‘𝐴)))
4536, 44sylbid 230 . . . . . . . . . . . . . . . 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 2962 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → ∀𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ≼ (ℵ‘𝐴))
49 iundom 9349 . . . . . . . . . . . . 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 694 . . . . . . . . . . . 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 692 . . . . . . . . . . 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 7994 . . . . . . . . . . 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 692 . . . . . . . . . 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 451 . . . . . . . . 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 1859 . . . . . . . 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 8890 . . . . . . . . . 10 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
58 alephon 8877 . . . . . . . . . . 11 (ℵ‘𝐴) ∈ On
59 infxpen 8822 . . . . . . . . . . 11 (((ℵ‘𝐴) ∈ On ∧ ω ⊆ (ℵ‘𝐴)) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
6058, 59mpan 705 . . . . . . . . . 10 (ω ⊆ (ℵ‘𝐴) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
6157, 60sylbi 207 . . . . . . . . 9 (𝐴 ∈ On → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
62 breq1 4647 . . . . . . . . . . . 12 (𝑧 = (cf‘(ℵ‘suc 𝐴)) → (𝑧 ≺ (ℵ‘suc 𝐴) ↔ (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴)))
6362, 41vtoclri 3278 . . . . . . . . . . 11 ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴))
64 alephsucdom 8887 . . . . . . . . . . 11 (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ≼ (ℵ‘𝐴) ↔ (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴)))
6563, 64syl5ibr 236 . . . . . . . . . 10 (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) ≼ (ℵ‘𝐴)))
66 fvex 6188 . . . . . . . . . . 11 (ℵ‘𝐴) ∈ V
6766xpdom1 8044 . . . . . . . . . 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 8000 . . . . . . . . . 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 451 . . . . . . . . 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 445 . . . . . . 7 ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
73 domtr 7994 . . . . . . 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 692 . . . . . 6 ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝐴))
75 domnsym 8071 . . . . . 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 450 . . . 4 (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)))
781, 77mt2d 131 . . 3 (𝐴 ∈ On → ¬ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))
79 cfon 9062 . . . . 5 (cf‘(ℵ‘suc 𝐴)) ∈ On
80 cfle 9061 . . . . . 6 (cf‘(ℵ‘suc 𝐴)) ⊆ (ℵ‘suc 𝐴)
81 onsseleq 5753 . . . . . 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 223 . . . . 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 707 . . . 4 ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) ∨ (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
8483ori 390 . . 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 9058 . . 3 (cf‘∅) = ∅
87 alephfnon 8873 . . . . . . . 8 ℵ Fn On
88 fndm 5978 . . . . . . . 8 (ℵ Fn On → dom ℵ = On)
8987, 88ax-mp 5 . . . . . . 7 dom ℵ = On
9089eleq2i 2691 . . . . . 6 (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
91 sucelon 7002 . . . . . 6 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
9290, 91bitr4i 267 . . . . 5 (suc 𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
93 ndmfv 6205 . . . . 5 (¬ suc 𝐴 ∈ dom ℵ → (ℵ‘suc 𝐴) = ∅)
9492, 93sylnbir 321 . . . 4 𝐴 ∈ On → (ℵ‘suc 𝐴) = ∅)
9594fveq2d 6182 . . 3 𝐴 ∈ On → (cf‘(ℵ‘suc 𝐴)) = (cf‘∅))
9686, 95, 943eqtr4a 2680 . 2 𝐴 ∈ On → (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
9785, 96pm2.61i 176 1 (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wo 383  wa 384   = wceq 1481  wex 1702  wcel 1988  wral 2909  wrex 2910  Vcvv 3195  wss 3567  c0 3907   ciun 4511   class class class wbr 4644   × cxp 5102  dom cdm 5104  Oncon0 5711  Lim wlim 5712  suc csuc 5713   Fn wfn 5871  wf 5872  1-1wf1 5873  cfv 5876  ωcom 7050  cen 7937  cdom 7938  csdm 7939  cardccrd 8746  cale 8747  cfccf 8748
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-inf2 8523  ax-ac2 9270
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rmo 2917  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-se 5064  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-map 7844  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-oi 8400  df-har 8448  df-card 8750  df-aleph 8751  df-cf 8752  df-acn 8753  df-ac 8924
This theorem is referenced by:  pwcfsdom  9390
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