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Theorem cardsucnn 10008
Description: The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 10007. (Contributed by NM, 7-Nov-2008.)
Assertion
Ref Expression
cardsucnn (𝐴 ∈ Ο‰ β†’ (cardβ€˜suc 𝐴) = suc (cardβ€˜π΄))

Proof of Theorem cardsucnn
StepHypRef Expression
1 peano2 7894 . . 3 (𝐴 ∈ Ο‰ β†’ suc 𝐴 ∈ Ο‰)
2 cardnn 9986 . . 3 (suc 𝐴 ∈ Ο‰ β†’ (cardβ€˜suc 𝐴) = suc 𝐴)
31, 2syl 17 . 2 (𝐴 ∈ Ο‰ β†’ (cardβ€˜suc 𝐴) = suc 𝐴)
4 cardnn 9986 . . 3 (𝐴 ∈ Ο‰ β†’ (cardβ€˜π΄) = 𝐴)
5 suceq 6430 . . 3 ((cardβ€˜π΄) = 𝐴 β†’ suc (cardβ€˜π΄) = suc 𝐴)
64, 5syl 17 . 2 (𝐴 ∈ Ο‰ β†’ suc (cardβ€˜π΄) = suc 𝐴)
73, 6eqtr4d 2768 1 (𝐴 ∈ Ο‰ β†’ (cardβ€˜suc 𝐴) = suc (cardβ€˜π΄))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  suc csuc 6366  β€˜cfv 6543  Ο‰com 7868  cardccrd 9958
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-om 7869  df-1o 8485  df-er 8723  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-card 9962
This theorem is referenced by: (None)
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