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Theorem snsssn 3654
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 3052 . . 3  |-  ( { A }  C_  { B } 
<-> 
A. x ( x  e.  { A }  ->  x  e.  { B } ) )
2 velsn 3510 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
3 velsn 3510 . . . . 5  |-  ( x  e.  { B }  <->  x  =  B )
42, 3imbi12i 238 . . . 4  |-  ( ( x  e.  { A }  ->  x  e.  { B } )  <->  ( x  =  A  ->  x  =  B ) )
54albii 1429 . . 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 2932 . . 3  |-  ( A  e.  _V  ->  ( A. x ( x  =  A  ->  x  =  B )  ->  A  =  B ) )
97, 8ax-mp 7 . 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 1312    = wceq 1314    e. wcel 1463   _Vcvv 2657    C_ wss 3037   {csn 3493
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-sbc 2879  df-in 3043  df-ss 3050  df-sn 3499
This theorem is referenced by: (None)
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