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Theorem cardsucnn 9144
 Description: The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 9143. (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 7364 . . 3 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2 cardnn 9122 . . 3 (suc 𝐴 ∈ ω → (card‘suc 𝐴) = suc 𝐴)
31, 2syl 17 . 2 (𝐴 ∈ ω → (card‘suc 𝐴) = suc 𝐴)
4 cardnn 9122 . . 3 (𝐴 ∈ ω → (card‘𝐴) = 𝐴)
5 suceq 6041 . . 3 ((card‘𝐴) = 𝐴 → suc (card‘𝐴) = suc 𝐴)
64, 5syl 17 . 2 (𝐴 ∈ ω → suc (card‘𝐴) = suc 𝐴)
73, 6eqtr4d 2816 1 (𝐴 ∈ ω → (card‘suc 𝐴) = suc (card‘𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1601   ∈ wcel 2106  suc csuc 5978  ‘cfv 6135  ωcom 7343  cardccrd 9094 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-om 7344  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-card 9098 This theorem is referenced by: (None)
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