Theorem peano3OLD 7874

Description: Obsolete version of peano3 7873 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 6433	. 2 ⊢ suc 𝐴 ≠ ∅
2	1	a1i 11	1 ⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅)

Syntax hints:   → wi 4   ∈ wcel 2144   ≠ wne 2959  ∅c0 4287  suc csuc 6350  ωcom 7848

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736  ax-nul 5258

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-v 3458  df-dif 3909  df-un 3911  df-nul 4288  df-sn 4585  df-suc 6354

This theorem is referenced by: (None)