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Theorem peano4 4218
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 𝜔 B 𝜔) → (suc A = suc BA = B))

Proof of Theorem peano4
StepHypRef Expression
1 suc11g 4190 1 ((A 𝜔 B 𝜔) → (suc A = suc BA = B))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1230   wcel 1376  suc csuc 4023  𝜔com 4211
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1318  ax-7 1319  ax-gen 1320  ax-ie1 1365  ax-ie2 1366  ax-8 1378  ax-10 1379  ax-11 1380  ax-i12 1381  ax-bnd 1382  ax-4 1383  ax-17 1402  ax-i9 1406  ax-ial 1411  ax-i5r 1412  ax-ext 2003  ax-setind 4175
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1233  df-nf 1332  df-sb 1629  df-clab 2008  df-cleq 2014  df-clel 2017  df-nfc 2148  df-ral 2286  df-v 2534  df-dif 2894  df-un 2896  df-sn 3329  df-pr 3330  df-suc 4028
This theorem is referenced by: (None)
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