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Theorem difprsn 3756
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
difprsn  |-  ( { A ,  B }  \  { A } ) 
C_  { B }

Proof of Theorem difprsn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . . 5  |-  x  e. 
_V
21elpr 3658 . . . 4  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
3 elsn 3655 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
43notbii 287 . . . 4  |-  ( -.  x  e.  { A } 
<->  -.  x  =  A )
5 biorf 394 . . . . 5  |-  ( -.  x  =  A  -> 
( x  =  B  <-> 
( x  =  A  \/  x  =  B ) ) )
65biimparc 473 . . . 4  |-  ( ( ( x  =  A  \/  x  =  B )  /\  -.  x  =  A )  ->  x  =  B )
72, 4, 6syl2anb 465 . . 3  |-  ( ( x  e.  { A ,  B }  /\  -.  x  e.  { A } )  ->  x  =  B )
8 eldif 3162 . . 3  |-  ( x  e.  ( { A ,  B }  \  { A } )  <->  ( x  e.  { A ,  B }  /\  -.  x  e. 
{ A } ) )
9 elsn 3655 . . 3  |-  ( x  e.  { B }  <->  x  =  B )
107, 8, 93imtr4i 257 . 2  |-  ( x  e.  ( { A ,  B }  \  { A } )  ->  x  e.  { B } )
1110ssriv 3184 1  |-  ( { A ,  B }  \  { A } ) 
C_  { B }
Colors of variables: wff set class
Syntax hints:   -. wn 3    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    \ cdif 3149    C_ wss 3152   {csn 3640   {cpr 3641
This theorem is referenced by:  itg11  19046  en2other2  27382  pmtrprfv  27396
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-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-sn 3646  df-pr 3647
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