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Theorem difprsnss 3963
Description: Removal of a singleton from an unordered pair. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
difprsnss  |-  ( { A ,  B }  \  { A } ) 
C_  { B }

Proof of Theorem difprsnss
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2968 . . . . 5  |-  x  e. 
_V
21elpr 3861 . . . 4  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
3 elsn 3858 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
43notbii 289 . . . 4  |-  ( -.  x  e.  { A } 
<->  -.  x  =  A )
5 biorf 396 . . . . 5  |-  ( -.  x  =  A  -> 
( x  =  B  <-> 
( x  =  A  \/  x  =  B ) ) )
65biimparc 475 . . . 4  |-  ( ( ( x  =  A  \/  x  =  B )  /\  -.  x  =  A )  ->  x  =  B )
72, 4, 6syl2anb 467 . . 3  |-  ( ( x  e.  { A ,  B }  /\  -.  x  e.  { A } )  ->  x  =  B )
8 eldif 3319 . . 3  |-  ( x  e.  ( { A ,  B }  \  { A } )  <->  ( x  e.  { A ,  B }  /\  -.  x  e. 
{ A } ) )
9 elsn 3858 . . 3  |-  ( x  e.  { B }  <->  x  =  B )
107, 8, 93imtr4i 259 . 2  |-  ( x  e.  ( { A ,  B }  \  { A } )  ->  x  e.  { B } )
1110ssriv 3341 1  |-  ( { A ,  B }  \  { A } ) 
C_  { B }
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 359    /\ wa 360    = wceq 1654    e. wcel 1728    \ cdif 3306    C_ wss 3309   {csn 3843   {cpr 3844
This theorem is referenced by:  itg11  19619  en2other2  27471  pmtrprfv  27485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-v 2967  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-sn 3849  df-pr 3850
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