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Theorem peano4 4566
Description: Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's 5 postulates for arithmetic. Proposition 7.30(4) of [TakeutiZaring] p. 43. (Contributed by NM, 3-Sep-2003.)
Assertion
Ref Expression
peano4  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )

Proof of Theorem peano4
StepHypRef Expression
1 nnon 4550 . 2  |-  ( A  e.  om  ->  A  e.  On )
2 nnon 4550 . 2  |-  ( B  e.  om  ->  B  e.  On )
3 suc11 4384 . 2  |-  ( ( A  e.  On  /\  B  e.  On )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
41, 2, 3syl2an 465 1  |-  ( ( A  e.  om  /\  B  e.  om )  ->  ( suc  A  =  suc  B  <->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1619    e. wcel 1621   Oncon0 4282   suc csuc 4284   omcom 4544
This theorem is referenced by:  dif1enOLD  6972  dif1en  6973  fseqdom  7534
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4035  ax-nul 4043  ax-pr 4105  ax-un 4400
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2511  df-rex 2512  df-rab 2514  df-v 2727  df-sbc 2920  df-dif 3078  df-un 3080  df-in 3082  df-ss 3086  df-pss 3088  df-nul 3360  df-if 3468  df-sn 3547  df-pr 3548  df-tp 3549  df-op 3550  df-uni 3725  df-br 3918  df-opab 3972  df-tr 4008  df-eprel 4195  df-po 4204  df-so 4205  df-fr 4242  df-we 4244  df-ord 4285  df-on 4286  df-lim 4287  df-suc 4288  df-om 4545
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