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Theorem alephreg 9283
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 8779 . . . 4 (𝐴 ∈ On → (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴))
2 alephon 8775 . . . . . . . . 9 (ℵ‘suc 𝐴) ∈ On
3 cff1 8963 . . . . . . . . 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 6113 . . . . . . . . . . . . 13 (cf‘(ℵ‘suc 𝐴)) ∈ V
6 fvex 6113 . . . . . . . . . . . . . 14 (𝑓𝑦) ∈ V
76sucex 6903 . . . . . . . . . . . . 13 suc (𝑓𝑦) ∈ V
85, 7iunex 7039 . . . . . . . . . . . 12 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ∈ V
9 f1f 6014 . . . . . . . . . . . . . 14 (𝑓:(cf‘(ℵ‘suc 𝐴))–1-1→(ℵ‘suc 𝐴) → 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴))
109ad2antrr 758 . . . . . . . . . . . . 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 788 . . . . . . . . . . . . 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 5752 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ (ℵ‘suc 𝐴) → 𝑥 ∈ On)
13 ffvelrn 6265 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑓𝑦) ∈ (ℵ‘suc 𝐴))
14 onelon 5665 . . . . . . . . . . . . . . . . . . . . . . 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 5730 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 ∈ On ∧ (𝑓𝑦) ∈ On) → (𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 ∈ suc (𝑓𝑦)))
1715, 16sylan2 490 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 ∈ On ∧ (𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴)))) → (𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 ∈ suc (𝑓𝑦)))
1817anassrs 678 . . . . . . . . . . . . . . . . . . . 20 (((𝑥 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → (𝑥 ⊆ (𝑓𝑦) ↔ 𝑥 ∈ suc (𝑓𝑦)))
1918rexbidva 3031 . . . . . . . . . . . . . . . . . . 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 4460 . . . . . . . . . . . . . . . . . . 19 (𝑥 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ↔ ∃𝑦 ∈ (cf‘(ℵ‘suc 𝐴))𝑥 ∈ suc (𝑓𝑦))
2119, 20syl6bbr 277 . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . 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 490 . . . . . . . . . . . . . . . 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 2968 . . . . . . . . . . . . . . 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 3558 . . . . . . . . . . . . . . 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 277 . . . . . . . . . . . . . 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 500 . . . . . . . . . . . . 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 691 . . . . . . . . . . . 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 7887 . . . . . . . . . . . 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 66 . . . . . . . . . . 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 790 . . . . . . . . . . . 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 6905 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ On → suc 𝐴 ∈ On)
33 alephislim 8789 . . . . . . . . . . . . . . . . . . 19 (suc 𝐴 ∈ On ↔ Lim (ℵ‘suc 𝐴))
34 limsuc 6941 . . . . . . . . . . . . . . . . . . 19 (Lim (ℵ‘suc 𝐴) → ((𝑓𝑦) ∈ (ℵ‘suc 𝐴) ↔ suc (𝑓𝑦) ∈ (ℵ‘suc 𝐴)))
3533, 34sylbi 206 . . . . . . . . . . . . . . . . . 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 4586 . . . . . . . . . . . . . . . . . . 19 (𝑧 = suc (𝑓𝑦) → (𝑧 ≺ (ℵ‘suc 𝐴) ↔ suc (𝑓𝑦) ≺ (ℵ‘suc 𝐴)))
38 alephcard 8776 . . . . . . . . . . . . . . . . . . . 20 (card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
39 iscard 8684 . . . . . . . . . . . . . . . . . . . . 21 ((card‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴) ↔ ((ℵ‘suc 𝐴) ∈ On ∧ ∀𝑧 ∈ (ℵ‘suc 𝐴)𝑧 ≺ (ℵ‘suc 𝐴)))
4039simprbi 479 . . . . . . . . . . . . . . . . . . . 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 3256 . . . . . . . . . . . . . . . . . 18 (suc (𝑓𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓𝑦) ≺ (ℵ‘suc 𝐴))
43 alephsucdom 8785 . . . . . . . . . . . . . . . . . 18 (𝐴 ∈ On → (suc (𝑓𝑦) ≼ (ℵ‘𝐴) ↔ suc (𝑓𝑦) ≺ (ℵ‘suc 𝐴)))
4442, 43syl5ibr 235 . . . . . . . . . . . . . . . . 17 (𝐴 ∈ On → (suc (𝑓𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓𝑦) ≼ (ℵ‘𝐴)))
4536, 44sylbid 229 . . . . . . . . . . . . . . . 16 (𝐴 ∈ On → ((𝑓𝑦) ∈ (ℵ‘suc 𝐴) → suc (𝑓𝑦) ≼ (ℵ‘𝐴)))
4613, 45syl5 33 . . . . . . . . . . . . . . 15 (𝐴 ∈ On → ((𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴) ∧ 𝑦 ∈ (cf‘(ℵ‘suc 𝐴))) → suc (𝑓𝑦) ≼ (ℵ‘𝐴)))
4746expdimp 452 . . . . . . . . . . . . . 14 ((𝐴 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → (𝑦 ∈ (cf‘(ℵ‘suc 𝐴)) → suc (𝑓𝑦) ≼ (ℵ‘𝐴)))
4847ralrimiv 2948 . . . . . . . . . . . . 13 ((𝐴 ∈ On ∧ 𝑓:(cf‘(ℵ‘suc 𝐴))⟶(ℵ‘suc 𝐴)) → ∀𝑦 ∈ (cf‘(ℵ‘suc 𝐴))suc (𝑓𝑦) ≼ (ℵ‘𝐴))
49 iundom 9243 . . . . . . . . . . . . 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 691 . . . . . . . . . . 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 7895 . . . . . . . . . . 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 691 . . . . . . . . . 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 450 . . . . . . . . 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 1848 . . . . . . . 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 8788 . . . . . . . . . 10 (𝐴 ∈ On ↔ ω ⊆ (ℵ‘𝐴))
58 alephon 8775 . . . . . . . . . . 11 (ℵ‘𝐴) ∈ On
59 infxpen 8720 . . . . . . . . . . 11 (((ℵ‘𝐴) ∈ On ∧ ω ⊆ (ℵ‘𝐴)) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
6058, 59mpan 702 . . . . . . . . . 10 (ω ⊆ (ℵ‘𝐴) → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
6157, 60sylbi 206 . . . . . . . . 9 (𝐴 ∈ On → ((ℵ‘𝐴) × (ℵ‘𝐴)) ≈ (ℵ‘𝐴))
62 breq1 4586 . . . . . . . . . . . 12 (𝑧 = (cf‘(ℵ‘suc 𝐴)) → (𝑧 ≺ (ℵ‘suc 𝐴) ↔ (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴)))
6362, 41vtoclri 3256 . . . . . . . . . . 11 ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴))
64 alephsucdom 8785 . . . . . . . . . . 11 (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ≼ (ℵ‘𝐴) ↔ (cf‘(ℵ‘suc 𝐴)) ≺ (ℵ‘suc 𝐴)))
6563, 64syl5ibr 235 . . . . . . . . . 10 (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → (cf‘(ℵ‘suc 𝐴)) ≼ (ℵ‘𝐴)))
66 fvex 6113 . . . . . . . . . . 11 (ℵ‘𝐴) ∈ V
6766xpdom1 7944 . . . . . . . . . 10 ((cf‘(ℵ‘suc 𝐴)) ≼ (ℵ‘𝐴) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴)))
6865, 67syl6 34 . . . . . . . . 9 (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ ((ℵ‘𝐴) × (ℵ‘𝐴))))
69 domentr 7901 . . . . . . . . . 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 450 . . . . . . . . 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 59 . . . . . . . 8 (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ (ℵ‘𝐴)))
7271imp 444 . . . . . . 7 ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → ((cf‘(ℵ‘suc 𝐴)) × (ℵ‘𝐴)) ≼ (ℵ‘𝐴))
73 domtr 7895 . . . . . . 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 691 . . . . . 6 ((𝐴 ∈ On ∧ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴)) → (ℵ‘suc 𝐴) ≼ (ℵ‘𝐴))
75 domnsym 7971 . . . . . 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 449 . . . 4 (𝐴 ∈ On → ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) → ¬ (ℵ‘𝐴) ≺ (ℵ‘suc 𝐴)))
781, 77mt2d 130 . . 3 (𝐴 ∈ On → ¬ (cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴))
79 cfon 8960 . . . . 5 (cf‘(ℵ‘suc 𝐴)) ∈ On
80 cfle 8959 . . . . . 6 (cf‘(ℵ‘suc 𝐴)) ⊆ (ℵ‘suc 𝐴)
81 onsseleq 5682 . . . . . 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 222 . . . . 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 704 . . . 4 ((cf‘(ℵ‘suc 𝐴)) ∈ (ℵ‘suc 𝐴) ∨ (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
8483ori 389 . . 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 8956 . . 3 (cf‘∅) = ∅
87 alephfnon 8771 . . . . . . . 8 ℵ Fn On
88 fndm 5904 . . . . . . . 8 (ℵ Fn On → dom ℵ = On)
8987, 88ax-mp 5 . . . . . . 7 dom ℵ = On
9089eleq2i 2680 . . . . . 6 (suc 𝐴 ∈ dom ℵ ↔ suc 𝐴 ∈ On)
91 sucelon 6909 . . . . . 6 (𝐴 ∈ On ↔ suc 𝐴 ∈ On)
9290, 91bitr4i 266 . . . . 5 (suc 𝐴 ∈ dom ℵ ↔ 𝐴 ∈ On)
93 ndmfv 6128 . . . . 5 (¬ suc 𝐴 ∈ dom ℵ → (ℵ‘suc 𝐴) = ∅)
9492, 93sylnbir 320 . . . 4 𝐴 ∈ On → (ℵ‘suc 𝐴) = ∅)
9594fveq2d 6107 . . 3 𝐴 ∈ On → (cf‘(ℵ‘suc 𝐴)) = (cf‘∅))
9686, 95, 943eqtr4a 2670 . 2 𝐴 ∈ On → (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴))
9785, 96pm2.61i 175 1 (cf‘(ℵ‘suc 𝐴)) = (ℵ‘suc 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 195  wo 382  wa 383   = wceq 1475  wex 1695  wcel 1977  wral 2896  wrex 2897  Vcvv 3173  wss 3540  c0 3874   ciun 4455   class class class wbr 4583   × cxp 5036  dom cdm 5038  Oncon0 5640  Lim wlim 5641  suc csuc 5642   Fn wfn 5799  wf 5800  1-1wf1 5801  cfv 5804  ωcom 6957  cen 7838  cdom 7839  csdm 7840  cardccrd 8644  cale 8645  cfccf 8646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-ac2 9168
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-oi 8298  df-har 8346  df-card 8648  df-aleph 8649  df-cf 8650  df-acn 8651  df-ac 8822
This theorem is referenced by:  pwcfsdom  9284
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