Theorem sneqrg 4839

Description: Closed form of sneqr 4840. (Contributed by Scott Fenton, 1-Apr-2011.) (Proof shortened by JJ, 23-Jul-2021.)

Assertion

Ref	Expression
sneqrg	⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))

Proof of Theorem sneqrg

Step	Hyp	Ref	Expression
1		snidg 4660	. . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴})
2		eleq2 2830	. . 3 ⊢ ({𝐴} = {𝐵} → (𝐴 ∈ {𝐴} ↔ 𝐴 ∈ {𝐵}))
3	1, 2	syl5ibcom 245	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 ∈ {𝐵}))
4		elsng 4640	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
5	3, 4	sylibd 239	1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  {csn 4626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-sn 4627

This theorem is referenced by:  sneqr  4840  sneqbg  4843  snsssng  32533  preimane  32680  altopth1  35966  altopth2  35967