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Theorem cardsucnn 8796
Description: The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 8795. (Contributed by NM, 7-Nov-2008.)
Assertion
Ref Expression
cardsucnn (𝐴 ∈ ω → (card‘suc 𝐴) = suc (card‘𝐴))

Proof of Theorem cardsucnn
StepHypRef Expression
1 peano2 7071 . . 3 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2 cardnn 8774 . . 3 (suc 𝐴 ∈ ω → (card‘suc 𝐴) = suc 𝐴)
31, 2syl 17 . 2 (𝐴 ∈ ω → (card‘suc 𝐴) = suc 𝐴)
4 cardnn 8774 . . 3 (𝐴 ∈ ω → (card‘𝐴) = 𝐴)
5 suceq 5778 . . 3 ((card‘𝐴) = 𝐴 → suc (card‘𝐴) = suc 𝐴)
64, 5syl 17 . 2 (𝐴 ∈ ω → suc (card‘𝐴) = suc 𝐴)
73, 6eqtr4d 2657 1 (𝐴 ∈ ω → (card‘suc 𝐴) = suc (card‘𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  suc csuc 5713  cfv 5876  ωcom 7050  cardccrd 8746
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-om 7051  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-fin 7944  df-card 8750
This theorem is referenced by: (None)
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