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Theorem peano4 4216
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 4189 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 1373   wcel 1375  suc csuc 4026  𝜔com 4209
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 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-setind 4174
This theorem depends on definitions:  df-bi 110  df-3an 877  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2287  df-v 2535  df-dif 2898  df-un 2900  df-sn 3333  df-pr 3334  df-suc 4031
This theorem is referenced by: (None)
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