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Theorem peano3 3141

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 ∈ ω → suc A ≠ ∅)

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Proof of Theorem peano3

Step	Hyp	Ref	Expression
1		nsuceq0 3043	. 2 ⊢ suc A ≠ ∅
2	1	a1i 8	1 ⊢ (A ∈ ω → suc A ≠ ∅)

Colors of variables: wff set class

Syntax hints:   → wi 3   ∈ wcel 955   ≠ wne 1577  ∅c0 2270  suc csuc 2940  ωcom 3121

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-nul 2700

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-nul 2271  df-sn 2402  df-pr 2403  df-suc 2944

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