Theorem sneqbg 3778

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 3777	. 2 |- ( A e. V -> ( { A } = { B } -> A = B ) )
2		sneq 3618	. 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 1364   e. wcel 2160   {csn 3607

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sn 3613

This theorem is referenced by:  infpwfidom  7228