Theorem peano3 4688

Description: The successor of any natural number is not zero. One of Peano's five postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42. (Contributed by NM, 3-Sep-2003.)

Assertion

Ref	Expression
peano3	|- ( A e. om -> suc A =/= (/) )

Proof of Theorem peano3

Step	Hyp	Ref	Expression
1		nsuceq0g 4509	1 |- ( A e. om -> suc A =/= (/) )

Syntax hints:   -> wi 4   e. wcel 2200   =/= wne 2400   (/)c0 3491   suc csuc 4456   omcom 4682

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-v 2801  df-dif 3199  df-un 3201  df-nul 3492  df-sn 3672  df-suc 4462

This theorem is referenced by:  nndceq0  4710  frecabcl  6545  frecsuclem  6552  nnsucsssuc  6638  php5  7019  findcard2  7051  findcard2s  7052  omp1eomlem  7261  ctmlemr  7275  nnsf  16371  peano4nninf  16372