Theorem peano3 4107

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 3891	. 2 |- suc A =/= (/)
2	1	a1i 8	1 |- (A e. om -> suc A =/= (/))

Syntax hints:   -> wi 3   e. wcel 1588   =/= wne 2266  (/)c0 3082  suc csuc 3813  omcom 4085

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1592  ax-gen 1593  ax-8 1594  ax-9 1595  ax-10 1596  ax-11 1597  ax-12 1598  ax-17 1605  ax-4 1608  ax-5o 1610  ax-6o 1613  ax-9o 1763  ax-10o 1781  ax-16 1854  ax-11o 1864  ax-ext 2123  ax-nul 3613

This theorem depends on definitions:  df-bi 220  df-or 338  df-an 339  df-ex 1616  df-sb 1816  df-clab 2129  df-cleq 2134  df-clel 2137  df-ne 2268  df-v 2540  df-dif 2830  df-un 2832  df-nul 3083  df-sn 3242  df-suc 3817

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