Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sneqbg GIF version

Theorem sneqbg 3697
 Description: Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)
Assertion
Ref Expression
sneqbg (𝐴𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))

Proof of Theorem sneqbg
StepHypRef Expression
1 sneqrg 3696 . 2 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
2 sneq 3542 . 2 (𝐴 = 𝐵 → {𝐴} = {𝐵})
31, 2impbid1 141 1 (𝐴𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   = wceq 1332   ∈ wcel 1481  {csn 3531 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sn 3537 This theorem is referenced by:  infpwfidom  7070
 Copyright terms: Public domain W3C validator