Theorem peano4 4595

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 4555	1 |- ( ( A e. om /\ B e. om ) -> ( suc A = suc B <-> A = B ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   suc csuc 4364   omcom 4588

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4535

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-dif 3131  df-un 3133  df-sn 3598  df-pr 3599  df-suc 4370

This theorem is referenced by:  frecabcl  6397  pwle2  14608