Theorem peano3 3114

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 3016	. 2 |- suc A =/= (/)
2	1	a1i 8	1 |- (A e. om -> suc A =/= (/))

Syntax hints:   -> wi 3   e. wcel 1105   =/= wne 1561  (/)c0 2251  suc csuc 2913  omcom 3094

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-nul 2678

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-v 1787  df-dif 2020  df-un 2021  df-nul 2252  df-sn 2383  df-pr 2384  df-suc 2917

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