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Theorem eupath2lem1 21737
Description: Lemma for eupath2 21740. (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 2504 . . 3  |-  ( (/)  =  if ( A  =  B ,  (/) ,  { A ,  B }
)  ->  ( U  e.  (/)  <->  U  e.  if ( A  =  B ,  (/) ,  { A ,  B } ) ) )
21bibi1d 312 . 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 2504 . . 3  |-  ( { A ,  B }  =  if ( A  =  B ,  (/) ,  { A ,  B }
)  ->  ( U  e.  { A ,  B } 
<->  U  e.  if ( A  =  B ,  (/)
,  { A ,  B } ) ) )
43bibi1d 312 . 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 3620 . . . 4  |-  -.  U  e.  (/)
65a1i 11 . . 3  |-  ( ( U  e.  V  /\  A  =  B )  ->  -.  U  e.  (/) )
7 simpl 445 . . . . 5  |-  ( ( A  =/=  B  /\  ( U  =  A  \/  U  =  B
) )  ->  A  =/=  B )
87neneqd 2624 . . . 4  |-  ( ( A  =/=  B  /\  ( U  =  A  \/  U  =  B
) )  ->  -.  A  =  B )
9 simpr 449 . . . 4  |-  ( ( U  e.  V  /\  A  =  B )  ->  A  =  B )
108, 9nsyl3 114 . . 3  |-  ( ( U  e.  V  /\  A  =  B )  ->  -.  ( A  =/= 
B  /\  ( U  =  A  \/  U  =  B ) ) )
116, 102falsed 342 . 2  |-  ( ( U  e.  V  /\  A  =  B )  ->  ( U  e.  (/)  <->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B )
) ) )
12 elprg 3860 . . 3  |-  ( U  e.  V  ->  ( U  e.  { A ,  B }  <->  ( U  =  A  \/  U  =  B ) ) )
13 df-ne 2608 . . . 4  |-  ( A  =/=  B  <->  -.  A  =  B )
14 ibar 492 . . . 4  |-  ( A  =/=  B  ->  (
( U  =  A  \/  U  =  B )  <->  ( A  =/= 
B  /\  ( U  =  A  \/  U  =  B ) ) ) )
1513, 14sylbir 206 . . 3  |-  ( -.  A  =  B  -> 
( ( U  =  A  \/  U  =  B )  <->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B ) ) ) )
1612, 15sylan9bb 682 . 2  |-  ( ( U  e.  V  /\  -.  A  =  B
)  ->  ( U  e.  { A ,  B } 
<->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B ) ) ) )
172, 4, 11, 16ifbothda 3796 1  |-  ( U  e.  V  ->  ( U  e.  if ( A  =  B ,  (/)
,  { A ,  B } )  <->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1654    e. wcel 1728    =/= wne 2606   (/)c0 3616   ifcif 3767   {cpr 3844
This theorem is referenced by:  eupath2lem2  21738  eupath2lem3  21739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-v 2967  df-dif 3312  df-un 3314  df-nul 3617  df-if 3768  df-sn 3849  df-pr 3850
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