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Theorem cardennn 9093
Description: If 𝐴 is equinumerous to a natural number, then that number is its cardinal. (Contributed by Mario Carneiro, 11-Jan-2013.)
Assertion
Ref Expression
cardennn ((𝐴𝐵𝐵 ∈ ω) → (card‘𝐴) = 𝐵)

Proof of Theorem cardennn
StepHypRef Expression
1 carden2b 9077 . 2 (𝐴𝐵 → (card‘𝐴) = (card‘𝐵))
2 cardnn 9073 . 2 (𝐵 ∈ ω → (card‘𝐵) = 𝐵)
31, 2sylan9eq 2851 1 ((𝐴𝐵𝐵 ∈ ω) → (card‘𝐴) = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157   class class class wbr 4841  cfv 6099  ωcom 7297  cen 8190  cardccrd 9045
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-pss 3783  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-tp 4371  df-op 4373  df-uni 4627  df-int 4666  df-br 4842  df-opab 4904  df-mpt 4921  df-tr 4944  df-id 5218  df-eprel 5223  df-po 5231  df-so 5232  df-fr 5269  df-we 5271  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-ord 5942  df-on 5943  df-lim 5944  df-suc 5945  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-om 7298  df-er 7980  df-en 8194  df-dom 8195  df-sdom 8196  df-fin 8197  df-card 9049
This theorem is referenced by:  dif1card  9117  fz1isolem  13490
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