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Theorem difprsnss 3658
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 2689 . . . . 5  |-  x  e. 
_V
21elpr 3548 . . . 4  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
3 velsn 3544 . . . . 5  |-  ( x  e.  { A }  <->  x  =  A )
43notbii 657 . . . 4  |-  ( -.  x  e.  { A } 
<->  -.  x  =  A )
5 biorf 733 . . . . 5  |-  ( -.  x  =  A  -> 
( x  =  B  <-> 
( x  =  A  \/  x  =  B ) ) )
65biimparc 297 . . . 4  |-  ( ( ( x  =  A  \/  x  =  B )  /\  -.  x  =  A )  ->  x  =  B )
72, 4, 6syl2anb 289 . . 3  |-  ( ( x  e.  { A ,  B }  /\  -.  x  e.  { A } )  ->  x  =  B )
8 eldif 3080 . . 3  |-  ( x  e.  ( { A ,  B }  \  { A } )  <->  ( x  e.  { A ,  B }  /\  -.  x  e. 
{ A } ) )
9 velsn 3544 . . 3  |-  ( x  e.  { B }  <->  x  =  B )
107, 8, 93imtr4i 200 . 2  |-  ( x  e.  ( { A ,  B }  \  { A } )  ->  x  e.  { B } )
1110ssriv 3101 1  |-  ( { A ,  B }  \  { A } ) 
C_  { B }
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    \/ wo 697    = wceq 1331    e. wcel 1480    \ cdif 3068    C_ wss 3071   {csn 3527   {cpr 3528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-sn 3533  df-pr 3534
This theorem is referenced by:  en2other2  7057
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