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Theorem eqoreldif 4004
Description: An element of a set is either equal to another element of the set or a member of the difference of the set and the singleton containing the other element. (Contributed by AV, 25-Aug-2020.)
Assertion
Ref Expression
eqoreldif  |-  ( B  e.  C  ->  ( A  e.  C  <->  ( A  =  B  \/  A  e.  ( C  \  { B } ) ) ) )

Proof of Theorem eqoreldif
StepHypRef Expression
1 orc 392 . . . . 5  |-  ( A  =  B  ->  ( A  =  B  \/  A  e.  ( C  \  { B } ) ) )
21a1d 25 . . . 4  |-  ( A  =  B  ->  (
( B  e.  C  /\  A  e.  C
)  ->  ( A  =  B  \/  A  e.  ( C  \  { B } ) ) ) )
3 simprr 774 . . . . . . 7  |-  ( ( -.  A  =  B  /\  ( B  e.  C  /\  A  e.  C ) )  ->  A  e.  C )
4 elsni 3985 . . . . . . . . . 10  |-  ( A  e.  { B }  ->  A  =  B )
54a1i 11 . . . . . . . . 9  |-  ( ( B  e.  C  /\  A  e.  C )  ->  ( A  e.  { B }  ->  A  =  B ) )
65con3d 140 . . . . . . . 8  |-  ( ( B  e.  C  /\  A  e.  C )  ->  ( -.  A  =  B  ->  -.  A  e.  { B } ) )
76impcom 437 . . . . . . 7  |-  ( ( -.  A  =  B  /\  ( B  e.  C  /\  A  e.  C ) )  ->  -.  A  e.  { B } )
83, 7eldifd 3401 . . . . . 6  |-  ( ( -.  A  =  B  /\  ( B  e.  C  /\  A  e.  C ) )  ->  A  e.  ( C  \  { B } ) )
98olcd 400 . . . . 5  |-  ( ( -.  A  =  B  /\  ( B  e.  C  /\  A  e.  C ) )  -> 
( A  =  B  \/  A  e.  ( C  \  { B } ) ) )
109ex 441 . . . 4  |-  ( -.  A  =  B  -> 
( ( B  e.  C  /\  A  e.  C )  ->  ( A  =  B  \/  A  e.  ( C  \  { B } ) ) ) )
112, 10pm2.61i 169 . . 3  |-  ( ( B  e.  C  /\  A  e.  C )  ->  ( A  =  B  \/  A  e.  ( C  \  { B } ) ) )
1211ex 441 . 2  |-  ( B  e.  C  ->  ( A  e.  C  ->  ( A  =  B  \/  A  e.  ( C  \  { B } ) ) ) )
13 eleq1 2537 . . . . 5  |-  ( A  =  B  ->  ( A  e.  C  <->  B  e.  C ) )
1413biimprd 231 . . . 4  |-  ( A  =  B  ->  ( B  e.  C  ->  A  e.  C ) )
15 eldifi 3544 . . . . 5  |-  ( A  e.  ( C  \  { B } )  ->  A  e.  C )
1615a1d 25 . . . 4  |-  ( A  e.  ( C  \  { B } )  -> 
( B  e.  C  ->  A  e.  C ) )
1714, 16jaoi 386 . . 3  |-  ( ( A  =  B  \/  A  e.  ( C  \  { B } ) )  ->  ( B  e.  C  ->  A  e.  C ) )
1817com12 31 . 2  |-  ( B  e.  C  ->  (
( A  =  B  \/  A  e.  ( C  \  { B } ) )  ->  A  e.  C )
)
1912, 18impbid 195 1  |-  ( B  e.  C  ->  ( A  e.  C  <->  ( A  =  B  \/  A  e.  ( C  \  { B } ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    \ cdif 3387   {csn 3959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-dif 3393  df-sn 3960
This theorem is referenced by:  lcmfunsnlem2  14692
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