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Theorem sneqbg 3875
 Description: Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)
Assertion
Ref Expression
sneqbg (A V → ({A} = {B} ↔ A = B))

Proof of Theorem sneqbg
StepHypRef Expression
1 sneqrg 3874 . 2 (A V → ({A} = {B} → A = B))
2 sneq 3744 . 2 (A = B → {A} = {B})
31, 2impbid1 194 1 (A V → ({A} = {B} ↔ A = B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710  {csn 3737 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sn 3741 This theorem is referenced by:  sneqb  3876
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