Theorem sneqbg 4794

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 4790	. 2 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} → 𝐴 = 𝐵))
2		sneq 4587	. 2 ⊢ (𝐴 = 𝐵 → {𝐴} = {𝐵})
3	1, 2	impbid1 225	1 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐵} ↔ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  {csn 4577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-sn 4578

This theorem is referenced by:  iotaval2  6453  suppval1  8099  suppsnop  8111  fseqdom  9920  infpwfidom  9922  canthwe  10545  s111  14522  initoid  17908  termoid  17909  embedsetcestrclem  18063  mat1dimelbas  22356  mat1dimbas  22357  unidifsnne  32485  altopthg  35961  altopthbg  35962  bj-snglc  36963  f1omptsnlem  37330  fvineqsnf1  37404  suceqsneq  38231  extid  38304  sn-iotalem  42214  eusnsn  47030