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Theorem snsssn 4150
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 4140 . 2  |-  ( { A }  C_  { B } 
<->  ( { A }  =  (/)  \/  { A }  =  { B } ) )
2 sneqr.1 . . . . . 6  |-  A  e. 
_V
32snnz 4102 . . . . 5  |-  { A }  =/=  (/)
43neii 2652 . . . 4  |-  -.  { A }  =  (/)
54pm2.21i 131 . . 3  |-  ( { A }  =  (/)  ->  A  =  B )
62sneqr 4149 . . 3  |-  ( { A }  =  { B }  ->  A  =  B )
75, 6jaoi 379 . 2  |-  ( ( { A }  =  (/) 
\/  { A }  =  { B } )  ->  A  =  B )
81, 7sylbi 195 1  |-  ( { A }  C_  { B }  ->  A  =  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    = wceq 1370    e. wcel 1758   _Vcvv 3078    C_ wss 3437   (/)c0 3746   {csn 3986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-v 3080  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3747  df-sn 3987
This theorem is referenced by: (None)
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