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Theorem difprsnss 4162
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 3116 . . . . 5  |-  x  e. 
_V
21elpr 4045 . . . 4  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
3 elsn 4041 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
43notbii 296 . . . 4  |-  ( -.  x  e.  { A } 
<->  -.  x  =  A )
5 biorf 405 . . . . 5  |-  ( -.  x  =  A  -> 
( x  =  B  <-> 
( x  =  A  \/  x  =  B ) ) )
65biimparc 487 . . . 4  |-  ( ( ( x  =  A  \/  x  =  B )  /\  -.  x  =  A )  ->  x  =  B )
72, 4, 6syl2anb 479 . . 3  |-  ( ( x  e.  { A ,  B }  /\  -.  x  e.  { A } )  ->  x  =  B )
8 eldif 3486 . . 3  |-  ( x  e.  ( { A ,  B }  \  { A } )  <->  ( x  e.  { A ,  B }  /\  -.  x  e. 
{ A } ) )
9 elsn 4041 . . 3  |-  ( x  e.  { B }  <->  x  =  B )
107, 8, 93imtr4i 266 . 2  |-  ( x  e.  ( { A ,  B }  \  { A } )  ->  x  e.  { B } )
1110ssriv 3508 1  |-  ( { A ,  B }  \  { A } ) 
C_  { B }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    \ cdif 3473    C_ wss 3476   {csn 4027   {cpr 4029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-sn 4028  df-pr 4030
This theorem is referenced by:  en2other2  8388  pmtrprfv  16293  itg11  21925
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