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Theorem alephreg 9364
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 8856 . . . 4 (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
2 alephon 8852 . . . . . . . . 9 (ℵ‘suc 𝐴) ∈ On
3 cff1 9040 . . . . . . . . 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 6168 . . . . . . . . . . . . 13 (cf‘(ℵ‘suc 𝐴)) ∈ V
6 fvex 6168 . . . . . . . . . . . . . 14 (𝑓𝑦) ∈ V
76sucex 6973 . . . . . . . . . . . . 13 suc (𝑓𝑦) ∈ V
85, 7iunex 7108 . . . . . . . . . . . 12 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ∈ V
9 f1f 6068 . . . . . . . . . . . . . 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 5804 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℵ‘suc 𝐴) → 𝑥 ∈ On)
13 ffvelrn 6323 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑓𝑦) ∈ (ℵ‘suc 𝐴))
14 onelon 5717 . . . . . . . . . . . . . . . . . . . . . . 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 5782 . . . . . . . . . . . . . . . . . . . . . 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 3044 . . . . . . . . . . . . . . . . . . 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 4497 . . . . . . . . . . . . . . . . . . 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 2981 . . . . . . . . . . . . . . 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 3578 . . . . . . . . . . . . . . 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 7961 . . . . . . . . . . . 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 6975 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ On → suc 𝐴 ∈ On)
33 alephislim 8866 . . . . . . . . . . . . . . . . . . 19 (suc 𝐴 ∈ On ↔ Lim (ℵ‘suc 𝐴))
34 limsuc 7011 . . . . . . . . . . . . . . . . . . 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 4626 . . . . . . . . . . . . . . . . . . 19 (𝑧 = suc (𝑓𝑦) → (𝑧 ≺ (ℵ‘suc 𝐴) ↔ suc (𝑓𝑦) ≺ (ℵ‘suc 𝐴)))
38 alephcard 8853 . . . . . . . . . . . . . . . . . . . 20 (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
39 iscard 8761 . . . . . . . . . . . . . . . . . . . . 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 3273 . . . . . . . . . . . . . . . . . 18 (suc (𝑓𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓𝑦) ≺ (ℵ‘suc 𝐴))
43 alephsucdom 8862 . . . . . . . . . . . . . . . . . 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 2961 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → ∀𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ≼ (ℵ‘𝐴))
49 iundom 9324 . . . . . . . . . . . . 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 7969 . . . . . . . . . . 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 1858 . . . . . . . 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 8865 . . . . . . . . . 10 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
58 alephon 8852 . . . . . . . . . . 11 (ℵ‘𝐴) ∈ On
59 infxpen 8797 . . . . . . . . . . 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 4626 . . . . . . . . . . . 12 (𝑧 = (cf‘(ℵ‘suc 𝐴)) → (𝑧 ≺ (ℵ‘suc 𝐴) ↔ (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴)))
6362, 41vtoclri 3273 . . . . . . . . . . 11 ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴))
64 alephsucdom 8862 . . . . . . . . . . 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 6168 . . . . . . . . . . 11 (ℵ‘𝐴) ∈ V
6766xpdom1 8019 . . . . . . . . . 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 7975 . . . . . . . . . 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 7969 . . . . . . 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 8046 . . . . . 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 9037 . . . . 5 (cf‘(ℵ‘suc 𝐴)) ∈ On
80 cfle 9036 . . . . . 6 (cf‘(ℵ‘suc 𝐴)) ⊆ (ℵ‘suc 𝐴)
81 onsseleq 5734 . . . . . 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 9033 . . 3 (cf‘∅) = ∅
87 alephfnon 8848 . . . . . . . 8 ℵ Fn On
88 fndm 5958 . . . . . . . 8 (ℵ Fn On → dom ℵ = On)
8987, 88ax-mp 5 . . . . . . 7 dom ℵ = On
9089eleq2i 2690 . . . . . 6 (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
91 sucelon 6979 . . . . . 6 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
9290, 91bitr4i 267 . . . . 5 (suc 𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
93 ndmfv 6185 . . . . 5 (¬ suc 𝐴 ∈ dom ℵ → (ℵ‘suc 𝐴) = ∅)
9492, 93sylnbir 321 . . . 4 𝐴 ∈ On → (ℵ‘suc 𝐴) = ∅)
9594fveq2d 6162 . . 3 𝐴 ∈ On → (cf‘(ℵ‘suc 𝐴)) = (cf‘∅))
9686, 95, 943eqtr4a 2681 . 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 1480  wex 1701  wcel 1987  wral 2908  wrex 2909  Vcvv 3190  wss 3560  c0 3897   ciun 4492   class class class wbr 4623   × cxp 5082  dom cdm 5084  Oncon0 5692  Lim wlim 5693  suc csuc 5694   Fn wfn 5852  wf 5853  1-1wf1 5854  cfv 5857  ωcom 7027  cen 7912  cdom 7913  csdm 7914  cardccrd 8721  cale 8722  cfccf 8723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-inf2 8498  ax-ac2 9245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-oadd 7524  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-oi 8375  df-har 8423  df-card 8725  df-aleph 8726  df-cf 8727  df-acn 8728  df-ac 8899
This theorem is referenced by:  pwcfsdom  9365
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