Theorem peano3 3157

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 3059	. 2 |- suc A =/= (/)
2	1	a1i 8	1 |- (A e. om -> suc A =/= (/))

Syntax hints:   -> wi 3   e. wcel 960   =/= wne 1588  (/)c0 2283  suc csuc 2956  omcom 3137

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-nul 2715

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-nul 2284  df-sn 2416  df-pr 2417  df-suc 2960

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