MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sneqrg Structured version   Visualization version   GIF version

Theorem sneqrg 4762
Description: Closed form of sneqr 4763. (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 4589 . . 3 (𝐴𝑉𝐴 ∈ {𝐴})
2 eleq2 2898 . . 3 ({𝐴} = {𝐵} → (𝐴 ∈ {𝐴} ↔ 𝐴 ∈ {𝐵}))
31, 2syl5ibcom 246 . 2 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 ∈ {𝐵}))
4 elsng 4571 . 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 1528  wcel 2105  {csn 4557
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-sn 4558
This theorem is referenced by:  sneqr  4763  sneqbg  4766  preimane  30343  altopth1  33323  altopth2  33324
  Copyright terms: Public domain W3C validator