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Theorem snsssn 3781
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
StepHypRef Expression
1 sssn 3772 . 2  |-  ( { A }  C_  { B } 
<->  ( { A }  =  (/)  \/  { A }  =  { B } ) )
2 sneqr.1 . . . . . 6  |-  A  e. 
_V
32snnz 3744 . . . . 5  |-  { A }  =/=  (/)
4 df-ne 2448 . . . . 5  |-  ( { A }  =/=  (/)  <->  -.  { A }  =  (/) )
53, 4mpbi 199 . . . 4  |-  -.  { A }  =  (/)
65pm2.21i 123 . . 3  |-  ( { A }  =  (/)  ->  A  =  B )
72sneqr 3780 . . 3  |-  ( { A }  =  { B }  ->  A  =  B )
86, 7jaoi 368 . 2  |-  ( ( { A }  =  (/) 
\/  { A }  =  { B } )  ->  A  =  B )
91, 8sylbi 187 1  |-  ( { A }  C_  { B }  ->  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 357    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788    C_ wss 3152   (/)c0 3455   {csn 3640
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646
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