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Theorem cardsucnn 9021
 Description: The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 9020. (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 7252 . . 3 (𝐴 ∈ ω → suc 𝐴 ∈ ω)
2 cardnn 8999 . . 3 (suc 𝐴 ∈ ω → (card‘suc 𝐴) = suc 𝐴)
31, 2syl 17 . 2 (𝐴 ∈ ω → (card‘suc 𝐴) = suc 𝐴)
4 cardnn 8999 . . 3 (𝐴 ∈ ω → (card‘𝐴) = 𝐴)
5 suceq 5951 . . 3 ((card‘𝐴) = 𝐴 → suc (card‘𝐴) = suc 𝐴)
64, 5syl 17 . 2 (𝐴 ∈ ω → suc (card‘𝐴) = suc 𝐴)
73, 6eqtr4d 2797 1 (𝐴 ∈ ω → (card‘suc 𝐴) = suc (card‘𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∈ wcel 2139  suc csuc 5886  ‘cfv 6049  ωcom 7231  cardccrd 8971 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-om 7232  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-card 8975 This theorem is referenced by: (None)
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