Theorem peano3 2392

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 2306	. 2 ⊢ ¬ suc A = ∅
2	1	a1i 7	1 ⊢ (A ∈ ω → ¬ suc A = ∅)

Syntax hints:  ¬ wn 1   → wi 2   = wceq 1091   ∈ wcel 1092  ∅c0 1707  suc csuc 2201  ωcom 2372

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-nul 1708  df-sn 1811  df-pr 1812  df-suc 2205

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