Theorem suceldifsn0 35788

Description: A successor that is a member of a set is also a member of that set with the empty set removed. (Contributed by BTernaryTau, 19-Oct-2026.)

Hypotheses

Ref	Expression
suceldifsn0.1	⊢ 𝐴 ∈ 𝐵
suceldifsn0.2	⊢ 𝐴 = suc 𝐶

Assertion

Ref	Expression
suceldifsn0	⊢ 𝐴 ∈ (𝐵 ∖ {∅})

Proof of Theorem suceldifsn0

Step	Hyp	Ref	Expression
1		suceldifsn0.1	. 2 ⊢ 𝐴 ∈ 𝐵
2		suceldifsn0.2	. . 3 ⊢ 𝐴 = suc 𝐶
3		nsuceq0 6463	. . 3 ⊢ suc 𝐶 ≠ ∅
4	2, 3	eqnetri 3029	. 2 ⊢ 𝐴 ≠ ∅
5		eldifsn 4748	. 2 ⊢ (𝐴 ∈ (𝐵 ∖ {∅}) ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ≠ ∅))
6	1, 4, 5	mpbir2an 724	1 ⊢ 𝐴 ∈ (𝐵 ∖ {∅})

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   ∖ cdif 4048  ∅c0 4281  {csn 4588  suc csuc 6379

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5263

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-v 3456  df-dif 4050  df-un 4154  df-nul 4282  df-sn 4589  df-suc 6383

This theorem is used by:  gdlopcfv1  35971  gdlopcfv2  35972  gdlopcfv3  35973  gdlopcfv4  35974  gdlopcfv5  35975  gdlopcfv6  35976  gdlopcfv7  35977  gdlopcfv8  35978