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Theorem alephon 8852
Description: An aleph is an ordinal number. (Contributed by NM, 10-Nov-2003.) (Revised by Mario Carneiro, 13-Sep-2013.)
Assertion
Ref Expression
alephon (ℵ‘𝐴) ∈ On

Proof of Theorem alephon
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 alephfnon 8848 . . 3 ℵ Fn On
2 fveq2 6158 . . . . . 6 (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
32eleq1d 2683 . . . . 5 (𝑥 = ∅ → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘∅) ∈ On))
4 fveq2 6158 . . . . . 6 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
54eleq1d 2683 . . . . 5 (𝑥 = 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘𝑦) ∈ On))
6 fveq2 6158 . . . . . 6 (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
76eleq1d 2683 . . . . 5 (𝑥 = suc 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘suc 𝑦) ∈ On))
8 aleph0 8849 . . . . . 6 (ℵ‘∅) = ω
9 omelon 8503 . . . . . 6 ω ∈ On
108, 9eqeltri 2694 . . . . 5 (ℵ‘∅) ∈ On
11 alephsuc 8851 . . . . . . 7 (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
12 harcl 8426 . . . . . . 7 (har‘(ℵ‘𝑦)) ∈ On
1311, 12syl6eqel 2706 . . . . . 6 (𝑦 ∈ On → (ℵ‘suc 𝑦) ∈ On)
1413a1d 25 . . . . 5 (𝑦 ∈ On → ((ℵ‘𝑦) ∈ On → (ℵ‘suc 𝑦) ∈ On))
15 vex 3193 . . . . . . 7 𝑥 ∈ V
16 iunon 7396 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (ℵ‘𝑦) ∈ On) → 𝑦𝑥 (ℵ‘𝑦) ∈ On)
1715, 16mpan 705 . . . . . 6 (∀𝑦𝑥 (ℵ‘𝑦) ∈ On → 𝑦𝑥 (ℵ‘𝑦) ∈ On)
18 alephlim 8850 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
1915, 18mpan 705 . . . . . . 7 (Lim 𝑥 → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2019eleq1d 2683 . . . . . 6 (Lim 𝑥 → ((ℵ‘𝑥) ∈ On ↔ 𝑦𝑥 (ℵ‘𝑦) ∈ On))
2117, 20syl5ibr 236 . . . . 5 (Lim 𝑥 → (∀𝑦𝑥 (ℵ‘𝑦) ∈ On → (ℵ‘𝑥) ∈ On))
223, 5, 7, 5, 10, 14, 21tfinds 7021 . . . 4 (𝑦 ∈ On → (ℵ‘𝑦) ∈ On)
2322rgen 2918 . . 3 𝑦 ∈ On (ℵ‘𝑦) ∈ On
24 ffnfv 6354 . . 3 (ℵ:On⟶On ↔ (ℵ Fn On ∧ ∀𝑦 ∈ On (ℵ‘𝑦) ∈ On))
251, 23, 24mpbir2an 954 . 2 ℵ:On⟶On
26 0elon 5747 . 2 ∅ ∈ On
2725, 26f0cli 6336 1 (ℵ‘𝐴) ∈ On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  c0 3897   ciun 4492  Oncon0 5692  Lim wlim 5693  suc csuc 5694   Fn wfn 5852  wf 5853  cfv 5857  ωcom 7027  harchar 8421  cale 8722
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
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-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-om 7028  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-en 7916  df-dom 7917  df-oi 8375  df-har 8423  df-aleph 8726
This theorem is referenced by:  alephnbtwn  8854  alephnbtwn2  8855  alephordilem1  8856  alephord  8858  alephord2  8859  alephord3  8861  alephsucdom  8862  alephsuc2  8863  alephf1  8868  alephsdom  8869  alephdom2  8870  alephle  8871  cardaleph  8872  alephf1ALT  8886  alephfp  8891  dfac12k  8929  alephsing  9058  alephval2  9354  alephadd  9359  alephmul  9360  alephexp1  9361  alephsuc3  9362  alephreg  9364  pwcfsdom  9365  cfpwsdom  9366  gchaleph  9453  gchaleph2  9454  gch2  9457
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