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Theorem snsssn 3695
Description: If a singleton is a subset of another, their members are equal. (Contributed by NM, 28-May-2006.)
Hypothesis
Ref Expression
sneqr.1  |-  A  e. 
_V
Assertion
Ref Expression
snsssn  |-  ( { A }  C_  { B }  ->  A  =  B )

Proof of Theorem snsssn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 dfss2 3090 . . 3  |-  ( { A }  C_  { B } 
<-> 
A. x ( x  e.  { A }  ->  x  e.  { B } ) )
2 velsn 3548 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
3 velsn 3548 . . . . 5  |-  ( x  e.  { B }  <->  x  =  B )
42, 3imbi12i 238 . . . 4  |-  ( ( x  e.  { A }  ->  x  e.  { B } )  <->  ( x  =  A  ->  x  =  B ) )
54albii 1447 . . 3  |-  ( A. x ( x  e. 
{ A }  ->  x  e.  { B }
)  <->  A. x ( x  =  A  ->  x  =  B ) )
61, 5bitri 183 . 2  |-  ( { A }  C_  { B } 
<-> 
A. x ( x  =  A  ->  x  =  B ) )
7 sneqr.1 . . 3  |-  A  e. 
_V
8 sbceqal 2967 . . 3  |-  ( A  e.  _V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  A  =  B ) )
97, 8ax-mp 5 . 2  |-  ( A. x ( x  =  A  ->  x  =  B )  ->  A  =  B )
106, 9sylbi 120 1  |-  ( { A }  C_  { B }  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1330    = wceq 1332    e. wcel 1481   _Vcvv 2689    C_ wss 3075   {csn 3531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2913  df-in 3081  df-ss 3088  df-sn 3537
This theorem is referenced by: (None)
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