MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  difsn Structured version   Unicode version

Theorem difsn 4145
Description: An element not in a set can be removed without affecting the set. (Contributed by NM, 16-Mar-2006.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
difsn  |-  ( -.  A  e.  B  -> 
( B  \  { A } )  =  B )

Proof of Theorem difsn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eldifsn 4136 . . 3  |-  ( x  e.  ( B  \  { A } )  <->  ( x  e.  B  /\  x  =/=  A ) )
2 simpl 457 . . . 4  |-  ( ( x  e.  B  /\  x  =/=  A )  ->  x  e.  B )
3 eleq1 2513 . . . . . . . 8  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
43biimpcd 224 . . . . . . 7  |-  ( x  e.  B  ->  (
x  =  A  ->  A  e.  B )
)
54necon3bd 2653 . . . . . 6  |-  ( x  e.  B  ->  ( -.  A  e.  B  ->  x  =/=  A ) )
65com12 31 . . . . 5  |-  ( -.  A  e.  B  -> 
( x  e.  B  ->  x  =/=  A ) )
76ancld 553 . . . 4  |-  ( -.  A  e.  B  -> 
( x  e.  B  ->  ( x  e.  B  /\  x  =/=  A
) ) )
82, 7impbid2 204 . . 3  |-  ( -.  A  e.  B  -> 
( ( x  e.  B  /\  x  =/= 
A )  <->  x  e.  B ) )
91, 8syl5bb 257 . 2  |-  ( -.  A  e.  B  -> 
( x  e.  ( B  \  { A } )  <->  x  e.  B ) )
109eqrdv 2438 1  |-  ( -.  A  e.  B  -> 
( B  \  { A } )  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1381    e. wcel 1802    =/= wne 2636    \ cdif 3455   {csn 4010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-v 3095  df-dif 3461  df-sn 4011
This theorem is referenced by:  difsnb  4153  difsnexi  6589  domdifsn  7598  domunsncan  7615  frfi  7763  infdifsn  8071  dfn2  10809  clslp  19515  nbgrassovt  24300  xrge00  27540
  Copyright terms: Public domain W3C validator