Theorem peano3 3147

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 ≠ ∅)

Proof of Theorem peano3

Step	Hyp	Ref	Expression
1		nsuceq0 3049	. 2 ⊢ suc A ≠ ∅
2	1	a1i 8	1 ⊢ (A ∈ ω → suc A ≠ ∅)

Syntax hints:   → wi 3   ∈ wcel 957   ≠ wne 1583  ∅c0 2277  suc csuc 2946  ωcom 3127

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-nul 2706

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-nul 2278  df-sn 2409  df-pr 2410  df-suc 2950

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