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Theorem sneqrg 4683
 Description: Closed form of sneqr 4684. (Contributed by Scott Fenton, 1-Apr-2011.) (Proof shortened by JJ, 23-Jul-2021.)
Assertion
Ref Expression
sneqrg (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))

Proof of Theorem sneqrg
StepHypRef Expression
1 snidg 4510 . . 3 (𝐴𝑉𝐴 ∈ {𝐴})
2 eleq2 2873 . . 3 ({𝐴} = {𝐵} → (𝐴 ∈ {𝐴} ↔ 𝐴 ∈ {𝐵}))
31, 2syl5ibcom 246 . 2 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 ∈ {𝐵}))
4 elsng 4492 . 2 (𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
53, 4sylibd 240 1 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1525   ∈ wcel 2083  {csn 4478 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-sn 4479 This theorem is referenced by:  sneqr  4684  sneqbg  4687  preimane  30101  altopth1  33037  altopth2  33038
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