Theorem peano4 4554

Description: Two natural numbers are equal iff their successors are equal, i.e. the successor function is one-to-one. One of Peano's five 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

Step	Hyp	Ref	Expression
1		suc11g 4514	1 |- ( ( A e. om /\ B e. om ) -> ( suc A = suc B <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   suc csuc 4324   omcom 4547

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139  ax-setind 4494

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714  df-dif 3104  df-un 3106  df-sn 3566  df-pr 3567  df-suc 4330

This theorem is referenced by:  frecabcl  6340  pwle2  13531