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Theorem oncardid 9115
Description: Any ordinal number is equinumerous to its cardinal number. Unlike cardid 9704, this theorem does not require the Axiom of Choice. (Contributed by NM, 26-Jul-2004.)
Assertion
Ref Expression
oncardid (𝐴 ∈ On → (card‘𝐴) ≈ 𝐴)

Proof of Theorem oncardid
StepHypRef Expression
1 onenon 9108 . 2 (𝐴 ∈ On → 𝐴 ∈ dom card)
2 cardid2 9112 . 2 (𝐴 ∈ dom card → (card‘𝐴) ≈ 𝐴)
31, 2syl 17 1 (𝐴 ∈ On → (card‘𝐴) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2106   class class class wbr 4886  dom cdm 5355  Oncon0 5976  cfv 6135  cen 8238  cardccrd 9094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ord 5979  df-on 5980  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-en 8242  df-card 9098
This theorem is referenced by:  cardom  9145  alephinit  9251  dfac12k  9304
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