Theorem peano3OLD 7835

Description: Obsolete version of peano3 7834 as of 10-Jun-2026. (Contributed by NM, 3-Sep-2003.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
peano3OLD	⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅)

Proof of Theorem peano3OLD

Step	Hyp	Ref	Expression
1		nsuceq0 6398	. 2 ⊢ suc 𝐴 ≠ ∅
2	1	a1i 11	1 ⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2121   ≠ wne 2936  ∅c0 4263  suc csuc 6315  ωcom 7809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-nul 5230

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ne 2937  df-v 3435  df-dif 3887  df-un 3889  df-nul 4264  df-sn 4558  df-suc 6319

This theorem is referenced by: (None)