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Theorem alephon 8930
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 8926 . . 3 ℵ Fn On
2 fveq2 6229 . . . . . 6 (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
32eleq1d 2715 . . . . 5 (𝑥 = ∅ → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘∅) ∈ On))
4 fveq2 6229 . . . . . 6 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
54eleq1d 2715 . . . . 5 (𝑥 = 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘𝑦) ∈ On))
6 fveq2 6229 . . . . . 6 (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
76eleq1d 2715 . . . . 5 (𝑥 = suc 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘suc 𝑦) ∈ On))
8 aleph0 8927 . . . . . 6 (ℵ‘∅) = ω
9 omelon 8581 . . . . . 6 ω ∈ On
108, 9eqeltri 2726 . . . . 5 (ℵ‘∅) ∈ On
11 alephsuc 8929 . . . . . . 7 (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
12 harcl 8507 . . . . . . 7 (har‘(ℵ‘𝑦)) ∈ On
1311, 12syl6eqel 2738 . . . . . 6 (𝑦 ∈ On → (ℵ‘suc 𝑦) ∈ On)
1413a1d 25 . . . . 5 (𝑦 ∈ On → ((ℵ‘𝑦) ∈ On → (ℵ‘suc 𝑦) ∈ On))
15 vex 3234 . . . . . . 7 𝑥 ∈ V
16 iunon 7481 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (ℵ‘𝑦) ∈ On) → 𝑦𝑥 (ℵ‘𝑦) ∈ On)
1715, 16mpan 706 . . . . . 6 (∀𝑦𝑥 (ℵ‘𝑦) ∈ On → 𝑦𝑥 (ℵ‘𝑦) ∈ On)
18 alephlim 8928 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
1915, 18mpan 706 . . . . . . 7 (Lim 𝑥 → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2019eleq1d 2715 . . . . . 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 7101 . . . 4 (𝑦 ∈ On → (ℵ‘𝑦) ∈ On)
2322rgen 2951 . . 3 𝑦 ∈ On (ℵ‘𝑦) ∈ On
24 ffnfv 6428 . . 3 (ℵ:On⟶On ↔ (ℵ Fn On ∧ ∀𝑦 ∈ On (ℵ‘𝑦) ∈ On))
251, 23, 24mpbir2an 975 . 2 ℵ:On⟶On
26 0elon 5816 . 2 ∅ ∈ On
2725, 26f0cli 6410 1 (ℵ‘𝐴) ∈ On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  wcel 2030  wral 2941  Vcvv 3231  c0 3948   ciun 4552  Oncon0 5761  Lim wlim 5762  suc csuc 5763   Fn wfn 5921  wf 5922  cfv 5926  ωcom 7107  harchar 8502  cale 8800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-isom 5935  df-riota 6651  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-en 7998  df-dom 7999  df-oi 8456  df-har 8504  df-aleph 8804
This theorem is referenced by:  alephnbtwn  8932  alephnbtwn2  8933  alephordilem1  8934  alephord  8936  alephord2  8937  alephord3  8939  alephsucdom  8940  alephsuc2  8941  alephf1  8946  alephsdom  8947  alephdom2  8948  alephle  8949  cardaleph  8950  alephf1ALT  8964  alephfp  8969  dfac12k  9007  alephsing  9136  alephval2  9432  alephadd  9437  alephmul  9438  alephexp1  9439  alephsuc3  9440  alephreg  9442  pwcfsdom  9443  cfpwsdom  9444  gchaleph  9531  gchaleph2  9532  gch2  9535
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