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Theorem cardsucnn 8418
Description: The cardinality of the successor of a finite ordinal (natural number). This theorem does not hold for infinite ordinals; see cardsucinf 8417. (Contributed by NM, 7-Nov-2008.)
Assertion
Ref Expression
cardsucnn  |-  ( A  e.  om  ->  ( card `  suc  A )  =  suc  ( card `  A ) )

Proof of Theorem cardsucnn
StepHypRef Expression
1 peano2 6727 . . 3  |-  ( A  e.  om  ->  suc  A  e.  om )
2 cardnn 8396 . . 3  |-  ( suc 
A  e.  om  ->  (
card `  suc  A )  =  suc  A )
31, 2syl 17 . 2  |-  ( A  e.  om  ->  ( card `  suc  A )  =  suc  A )
4 cardnn 8396 . . 3  |-  ( A  e.  om  ->  ( card `  A )  =  A )
5 suceq 5507 . . 3  |-  ( (
card `  A )  =  A  ->  suc  ( card `  A )  =  suc  A )
64, 5syl 17 . 2  |-  ( A  e.  om  ->  suc  ( card `  A )  =  suc  A )
73, 6eqtr4d 2473 1  |-  ( A  e.  om  ->  ( card `  suc  A )  =  suc  ( card `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1437    e. wcel 1870   suc csuc 5444   ` cfv 5601   omcom 6706   cardccrd 8368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-om 6707  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-card 8372
This theorem is referenced by: (None)
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