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

Theorem sneqrg 4402
Description: Closed form of sneqr 4403. (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 4239 . . 3 (𝐴𝑉𝐴 ∈ {𝐴})
2 eleq2 2719 . . 3 ({𝐴} = {𝐵} → (𝐴 ∈ {𝐴} ↔ 𝐴 ∈ {𝐵}))
31, 2syl5ibcom 235 . 2 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 ∈ {𝐵}))
4 elsng 4224 . 2 (𝐴𝑉 → (𝐴 ∈ {𝐵} ↔ 𝐴 = 𝐵))
53, 4sylibd 229 1 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  {csn 4210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sn 4211
This theorem is referenced by:  sneqr  4403  sneqbg  4406  altopth1  32197  altopth2  32198
  Copyright terms: Public domain W3C validator