Theorem sneqbg 3759

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 3758	. 2 |- ( A e. V -> ( { A } = { B } -> A = B ) )
2		sneq 3600	. 2 |- ( A = B -> { A } = { B } )
3	1, 2	impbid1 142	1 |- ( A e. V -> ( { A } = { B } <-> A = B ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1353   e. wcel 2146   {csn 3589

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-sn 3595

This theorem is referenced by:  infpwfidom  7187