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Theorem sneqbg 4560
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 4556 . 2 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
2 sneq 4378 . 2 (𝐴 = 𝐵 → {𝐴} = {𝐵})
31, 2impbid1 217 1 (𝐴𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198   = wceq 1653  wcel 2157  {csn 4368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-sn 4369
This theorem is referenced by:  suppval1  7538  suppsnop  7546  fseqdom  9135  infpwfidom  9137  canthwe  9761  s111  13635  initoid  16969  termoid  16970  embedsetcestrclem  17112  mat1dimelbas  20603  mat1dimbas  20604  altopthg  32587  altopthbg  32588  bj-snglc  33449  f1omptsnlem  33682  extid  34576  eusnsn  41914
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