Theorem peano3 3784

Description: The successor of any natural number is not zero. One of Peano's 5 postulates for arithmetic. Proposition 7.30(3) of [TakeutiZaring] p. 42.

Assertion

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

Proof of Theorem peano3

Step	Hyp	Ref	Expression
1		nsuceq0 3561	. 2 |- suc A =/= (/)
2	1	a1i 8	1 |- (A e. om -> suc A =/= (/))

Syntax hints:   -> wi 3   e. wcel 1138   =/= wne 1854  (/)c0 2701  suc csuc 3474  omcom 3760

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1142  ax-gen 1143  ax-8 1144  ax-9 1145  ax-10 1146  ax-11 1147  ax-12 1148  ax-17 1155  ax-4 1157  ax-5o 1159  ax-6o 1162  ax-9o 1319  ax-10o 1338  ax-16 1418  ax-11o 1426  ax-ext 1702  ax-nul 3260

This theorem depends on definitions:  df-bi 163  df-or 240  df-an 241  df-ex 1165  df-sb 1374  df-clab 1709  df-cleq 1714  df-clel 1717  df-ne 1856  df-v 2127  df-dif 2430  df-un 2433  df-nul 2702  df-sn 2873  df-suc 3478

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