Theorem peano3OLD 7897

Description: Obsolete version of peano3 7896 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 6453	. 2 ⊢ suc 𝐴 ≠ ∅
2	1	a1i 11	1 ⊢ (𝐴 ∈ ω → suc 𝐴 ≠ ∅)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2146   ≠ wne 2961  ∅c0 4289  suc csuc 6369  ωcom 7871

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-nul 5274

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-v 3460  df-dif 3911  df-un 3913  df-nul 4290  df-sn 4595  df-suc 6373

This theorem is used by: (None)