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Theorem cardsucnn 9982
Description: The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 9981. (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 7878 . . 3 (𝐴 ∈ Ο‰ β†’ suc 𝐴 ∈ Ο‰)
2 cardnn 9960 . . 3 (suc 𝐴 ∈ Ο‰ β†’ (cardβ€˜suc 𝐴) = suc 𝐴)
31, 2syl 17 . 2 (𝐴 ∈ Ο‰ β†’ (cardβ€˜suc 𝐴) = suc 𝐴)
4 cardnn 9960 . . 3 (𝐴 ∈ Ο‰ β†’ (cardβ€˜π΄) = 𝐴)
5 suceq 6424 . . 3 ((cardβ€˜π΄) = 𝐴 β†’ suc (cardβ€˜π΄) = suc 𝐴)
64, 5syl 17 . 2 (𝐴 ∈ Ο‰ β†’ suc (cardβ€˜π΄) = suc 𝐴)
73, 6eqtr4d 2769 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 6360  β€˜cfv 6537  Ο‰com 7852  cardccrd 9932
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-om 7853  df-1o 8467  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936
This theorem is referenced by: (None)
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