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Theorem sneqbg 4773
 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 4769 . 2 (𝐴𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
2 sneq 4576 . 2 (𝐴 = 𝐵 → {𝐴} = {𝐵})
31, 2impbid1 227 1 (𝐴𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   = wceq 1533   ∈ wcel 2110  {csn 4566 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-sn 4567 This theorem is referenced by:  suppval1  7835  suppsnop  7843  fseqdom  9451  infpwfidom  9453  canthwe  10072  s111  13968  initoid  17264  termoid  17265  embedsetcestrclem  17406  mat1dimelbas  21079  mat1dimbas  21080  unidifsnne  30295  altopthg  33428  altopthbg  33429  bj-snglc  34281  f1omptsnlem  34616  fvineqsnf1  34690  extid  35567  eusnsn  43262
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