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Theorem difprsnss 4107
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 3048 . . . . 5  |-  x  e. 
_V
21elpr 3986 . . . 4  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
3 elsn 3982 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
43notbii 298 . . . 4  |-  ( -.  x  e.  { A } 
<->  -.  x  =  A )
5 biorf 407 . . . . 5  |-  ( -.  x  =  A  -> 
( x  =  B  <-> 
( x  =  A  \/  x  =  B ) ) )
65biimparc 490 . . . 4  |-  ( ( ( x  =  A  \/  x  =  B )  /\  -.  x  =  A )  ->  x  =  B )
72, 4, 6syl2anb 482 . . 3  |-  ( ( x  e.  { A ,  B }  /\  -.  x  e.  { A } )  ->  x  =  B )
8 eldif 3414 . . 3  |-  ( x  e.  ( { A ,  B }  \  { A } )  <->  ( x  e.  { A ,  B }  /\  -.  x  e. 
{ A } ) )
9 elsn 3982 . . 3  |-  ( x  e.  { B }  <->  x  =  B )
107, 8, 93imtr4i 270 . 2  |-  ( x  e.  ( { A ,  B }  \  { A } )  ->  x  e.  { B } )
1110ssriv 3436 1  |-  ( { A ,  B }  \  { A } ) 
C_  { B }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 370    /\ wa 371    = wceq 1444    e. wcel 1887    \ cdif 3401    C_ wss 3404   {csn 3968   {cpr 3970
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-sn 3969  df-pr 3971
This theorem is referenced by:  en2other2  8440  pmtrprfv  17094  itg11  22649
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