Theorem peano3 3151

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 3053	. 2 |- suc A =/= (/)
2	1	a1i 8	1 |- (A e. om -> suc A =/= (/))

Syntax hints:   -> wi 3   e. wcel 958   =/= wne 1585  (/)c0 2280  suc csuc 2950  omcom 3131

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-nul 2710

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-nul 2281  df-sn 2412  df-pr 2413  df-suc 2954

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