Theorem sneqbg 3883

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 3882	. 2 |- ( A e. V -> ( { A } = { B } -> A = B ) )
2		sneq 3716	. 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 1402   e. wcel 2209   {csn 3705

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sn 3711

This theorem is referenced by:  suppval1  6469  suppsnopdc  6480  eqsndc  7200  infpwfidom  7540  s111  11377