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Theorem alephon 9473
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 9469 . . 3 ℵ Fn On
2 fveq2 6646 . . . . . 6 (𝑥 = ∅ → (ℵ‘𝑥) = (ℵ‘∅))
32eleq1d 2895 . . . . 5 (𝑥 = ∅ → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘∅) ∈ On))
4 fveq2 6646 . . . . . 6 (𝑥 = 𝑦 → (ℵ‘𝑥) = (ℵ‘𝑦))
54eleq1d 2895 . . . . 5 (𝑥 = 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘𝑦) ∈ On))
6 fveq2 6646 . . . . . 6 (𝑥 = suc 𝑦 → (ℵ‘𝑥) = (ℵ‘suc 𝑦))
76eleq1d 2895 . . . . 5 (𝑥 = suc 𝑦 → ((ℵ‘𝑥) ∈ On ↔ (ℵ‘suc 𝑦) ∈ On))
8 aleph0 9470 . . . . . 6 (ℵ‘∅) = ω
9 omelon 9087 . . . . . 6 ω ∈ On
108, 9eqeltri 2907 . . . . 5 (ℵ‘∅) ∈ On
11 alephsuc 9472 . . . . . . 7 (𝑦 ∈ On → (ℵ‘suc 𝑦) = (har‘(ℵ‘𝑦)))
12 harcl 9003 . . . . . . 7 (har‘(ℵ‘𝑦)) ∈ On
1311, 12eqeltrdi 2919 . . . . . 6 (𝑦 ∈ On → (ℵ‘suc 𝑦) ∈ On)
1413a1d 25 . . . . 5 (𝑦 ∈ On → ((ℵ‘𝑦) ∈ On → (ℵ‘suc 𝑦) ∈ On))
15 vex 3476 . . . . . . 7 𝑥 ∈ V
16 iunon 7954 . . . . . . 7 ((𝑥 ∈ V ∧ ∀𝑦𝑥 (ℵ‘𝑦) ∈ On) → 𝑦𝑥 (ℵ‘𝑦) ∈ On)
1715, 16mpan 688 . . . . . 6 (∀𝑦𝑥 (ℵ‘𝑦) ∈ On → 𝑦𝑥 (ℵ‘𝑦) ∈ On)
18 alephlim 9471 . . . . . . . 8 ((𝑥 ∈ V ∧ Lim 𝑥) → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
1915, 18mpan 688 . . . . . . 7 (Lim 𝑥 → (ℵ‘𝑥) = 𝑦𝑥 (ℵ‘𝑦))
2019eleq1d 2895 . . . . . 6 (Lim 𝑥 → ((ℵ‘𝑥) ∈ On ↔ 𝑦𝑥 (ℵ‘𝑦) ∈ On))
2117, 20syl5ibr 248 . . . . 5 (Lim 𝑥 → (∀𝑦𝑥 (ℵ‘𝑦) ∈ On → (ℵ‘𝑥) ∈ On))
223, 5, 7, 5, 10, 14, 21tfinds 7552 . . . 4 (𝑦 ∈ On → (ℵ‘𝑦) ∈ On)
2322rgen 3135 . . 3 𝑦 ∈ On (ℵ‘𝑦) ∈ On
24 ffnfv 6858 . . 3 (ℵ:On⟶On ↔ (ℵ Fn On ∧ ∀𝑦 ∈ On (ℵ‘𝑦) ∈ On))
251, 23, 24mpbir2an 709 . 2 ℵ:On⟶On
26 0elon 6220 . 2 ∅ ∈ On
2725, 26f0cli 6840 1 (ℵ‘𝐴) ∈ On
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2114  wral 3125  Vcvv 3473  c0 4269   ciun 4895  Oncon0 6167  Lim wlim 6168  suc csuc 6169   Fn wfn 6326  wf 6327  cfv 6331  ωcom 7558  harchar 8998  cale 9343
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-inf2 9082
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-se 5491  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-isom 6340  df-riota 7091  df-om 7559  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-en 8488  df-dom 8489  df-oi 8952  df-har 9000  df-aleph 9347
This theorem is referenced by:  alephnbtwn  9475  alephnbtwn2  9476  alephordilem1  9477  alephord  9479  alephord2  9480  alephord3  9482  alephsucdom  9483  alephsuc2  9484  alephf1  9489  alephsdom  9490  alephdom2  9491  alephle  9492  cardaleph  9493  alephf1ALT  9507  alephfp  9512  dfac12k  9551  alephsing  9676  alephval2  9972  alephadd  9977  alephmul  9978  alephexp1  9979  alephsuc3  9980  alephreg  9982  pwcfsdom  9983  cfpwsdom  9984  gchaleph  10071  gchaleph2  10072  gch2  10075  alephiso2  40052
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