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Theorem difprsnss 4167
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 3112 . . . . 5  |-  x  e. 
_V
21elpr 4050 . . . 4  |-  ( x  e.  { A ,  B }  <->  ( x  =  A  \/  x  =  B ) )
3 elsn 4046 . . . . 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 3481 . . 3  |-  ( x  e.  ( { A ,  B }  \  { A } )  <->  ( x  e.  { A ,  B }  /\  -.  x  e. 
{ A } ) )
9 elsn 4046 . . 3  |-  ( x  e.  { B }  <->  x  =  B )
107, 8, 93imtr4i 266 . 2  |-  ( x  e.  ( { A ,  B }  \  { A } )  ->  x  e.  { B } )
1110ssriv 3503 1  |-  ( { A ,  B }  \  { A } ) 
C_  { B }
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    \/ wo 368    /\ wa 369    = wceq 1395    e. wcel 1819    \ cdif 3468    C_ wss 3471   {csn 4032   {cpr 4034
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-sn 4033  df-pr 4035
This theorem is referenced by:  en2other2  8404  pmtrprfv  16605  itg11  22224
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