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Theorem eupath2lem1 25550
Description: Lemma for eupath2 25553. (Contributed by Mario Carneiro, 8-Apr-2015.)
Assertion
Ref Expression
eupath2lem1  |-  ( U  e.  V  ->  ( U  e.  if ( A  =  B ,  (/)
,  { A ,  B } )  <->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B ) ) ) )

Proof of Theorem eupath2lem1
StepHypRef Expression
1 eleq2 2502 . . 3  |-  ( (/)  =  if ( A  =  B ,  (/) ,  { A ,  B }
)  ->  ( U  e.  (/)  <->  U  e.  if ( A  =  B ,  (/) ,  { A ,  B } ) ) )
21bibi1d 320 . 2  |-  ( (/)  =  if ( A  =  B ,  (/) ,  { A ,  B }
)  ->  ( ( U  e.  (/)  <->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B ) ) )  <-> 
( U  e.  if ( A  =  B ,  (/) ,  { A ,  B } )  <->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B ) ) ) ) )
3 eleq2 2502 . . 3  |-  ( { A ,  B }  =  if ( A  =  B ,  (/) ,  { A ,  B }
)  ->  ( U  e.  { A ,  B } 
<->  U  e.  if ( A  =  B ,  (/)
,  { A ,  B } ) ) )
43bibi1d 320 . 2  |-  ( { A ,  B }  =  if ( A  =  B ,  (/) ,  { A ,  B }
)  ->  ( ( U  e.  { A ,  B }  <->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B ) ) )  <-> 
( U  e.  if ( A  =  B ,  (/) ,  { A ,  B } )  <->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B ) ) ) ) )
5 noel 3771 . . . 4  |-  -.  U  e.  (/)
65a1i 11 . . 3  |-  ( ( U  e.  V  /\  A  =  B )  ->  -.  U  e.  (/) )
7 simpl 458 . . . . 5  |-  ( ( A  =/=  B  /\  ( U  =  A  \/  U  =  B
) )  ->  A  =/=  B )
87neneqd 2632 . . . 4  |-  ( ( A  =/=  B  /\  ( U  =  A  \/  U  =  B
) )  ->  -.  A  =  B )
9 simpr 462 . . . 4  |-  ( ( U  e.  V  /\  A  =  B )  ->  A  =  B )
108, 9nsyl3 122 . . 3  |-  ( ( U  e.  V  /\  A  =  B )  ->  -.  ( A  =/= 
B  /\  ( U  =  A  \/  U  =  B ) ) )
116, 102falsed 352 . 2  |-  ( ( U  e.  V  /\  A  =  B )  ->  ( U  e.  (/)  <->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B )
) ) )
12 elprg 4018 . . 3  |-  ( U  e.  V  ->  ( U  e.  { A ,  B }  <->  ( U  =  A  \/  U  =  B ) ) )
13 df-ne 2627 . . . 4  |-  ( A  =/=  B  <->  -.  A  =  B )
14 ibar 506 . . . 4  |-  ( A  =/=  B  ->  (
( U  =  A  \/  U  =  B )  <->  ( A  =/= 
B  /\  ( U  =  A  \/  U  =  B ) ) ) )
1513, 14sylbir 216 . . 3  |-  ( -.  A  =  B  -> 
( ( U  =  A  \/  U  =  B )  <->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B ) ) ) )
1612, 15sylan9bb 704 . 2  |-  ( ( U  e.  V  /\  -.  A  =  B
)  ->  ( U  e.  { A ,  B } 
<->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B ) ) ) )
172, 4, 11, 16ifbothda 3950 1  |-  ( U  e.  V  ->  ( U  e.  if ( A  =  B ,  (/)
,  { A ,  B } )  <->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1870    =/= wne 2625   (/)c0 3767   ifcif 3915   {cpr 4004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-v 3089  df-dif 3445  df-un 3447  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005
This theorem is referenced by:  eupath2lem2  25551  eupath2lem3  25552
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