Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  oncardid Structured version   Visualization version   GIF version

Theorem oncardid 9379
 Description: Any ordinal number is equinumerous to its cardinal number. Unlike cardid 9963, 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 9372 . 2 (𝐴 ∈ On → 𝐴 ∈ dom card)
2 cardid2 9376 . 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 2110   class class class wbr 5058  dom cdm 5549  Oncon0 6185  ‘cfv 6349   ≈ cen 8500  cardccrd 9358 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-int 4869  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-ord 6188  df-on 6189  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-en 8504  df-card 9362 This theorem is referenced by:  cardom  9409  alephinit  9515  dfac12k  9567  harval3  39897
 Copyright terms: Public domain W3C validator