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Theorem sneqbg 3562
 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 3561 . 2 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
2 sneq 3414 . 2 (𝐴 = 𝐵 → {𝐴} = {𝐵})
31, 2impbid1 134 1 (𝐴𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 102   = wceq 1259   ∈ wcel 1409  {csn 3403 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-sn 3409 This theorem is referenced by: (None)
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