Theorem sneqbg 4771

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

Step	Hyp	Ref	Expression
1		sneqrg 4767	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
2		sneq 4568	. 2 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
3	1, 2	impbid1 224	1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108  {csn 4558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-sn 4559

This theorem is referenced by:  suppval1  7954  suppsnop  7965  fseqdom  9713  infpwfidom  9715  canthwe  10338  s111  14248  initoid  17632  termoid  17633  embedsetcestrclem  17790  mat1dimelbas  21528  mat1dimbas  21529  unidifsnne  30785  altopthg  34196  altopthbg  34197  bj-snglc  35086  f1omptsnlem  35434  fvineqsnf1  35508  extid  36373  sn-iotalem  40117  sn-iotaval  40119  eusnsn  44407