Theorem sneqbg 3804

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 3803	. 2 |- ( A e. V -> ( { A } = { B } -> A = B ) )
2		sneq 3644	. 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 1373   e. wcel 2176   {csn 3633

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-sn 3639

This theorem is referenced by:  infpwfidom  7308  s111  11088