Theorem sneqbg 4374

Description: Two singletons of sets are equal iff their elements are equal. (Contributed by Scott Fenton, 16-Apr-2012.)

Assertion

Ref	Expression
sneqbg	|- ( A e. V -> ( { A } = { B } <-> A = B ) )

Proof of Theorem sneqbg

Step	Hyp	Ref	Expression
1		sneqrg 4370	. 2 |- ( A e. V -> ( { A } = { B } -> A = B ) )
2		sneq 4187	. 2 |- ( A = B -> { A } = { B } )
3	1, 2	impbid1 215	1 |- ( A e. V -> ( { A } = { B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   e. wcel 1990   {csn 4177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sn 4178

This theorem is referenced by:  suppval1  7301  suppsnop  7309  fseqdom  8849  infpwfidom  8851  canthwe  9473  s111  13395  initoid  16655  termoid  16656  embedsetcestrclem  16797  mat1dimelbas  20277  mat1dimbas  20278  altopthg  32074  altopthbg  32075  bj-snglc  32957  f1omptsnlem  33183  extid  34081  opideq  34110