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Theorem eupath2lem1 23916
Description: Lemma for eupath2 23919. (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 2357 . . 3  |-  ( (/)  =  if ( A  =  B ,  (/) ,  { A ,  B }
)  ->  ( U  e.  (/)  <->  U  e.  if ( A  =  B ,  (/) ,  { A ,  B } ) ) )
21bibi1d 310 . 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 2357 . . 3  |-  ( { A ,  B }  =  if ( A  =  B ,  (/) ,  { A ,  B }
)  ->  ( U  e.  { A ,  B } 
<->  U  e.  if ( A  =  B ,  (/)
,  { A ,  B } ) ) )
43bibi1d 310 . 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 3472 . . . 4  |-  -.  U  e.  (/)
65a1i 10 . . 3  |-  ( ( U  e.  V  /\  A  =  B )  ->  -.  U  e.  (/) )
7 simpl 443 . . . . 5  |-  ( ( A  =/=  B  /\  ( U  =  A  \/  U  =  B
) )  ->  A  =/=  B )
87neneqd 2475 . . . 4  |-  ( ( A  =/=  B  /\  ( U  =  A  \/  U  =  B
) )  ->  -.  A  =  B )
9 simpr 447 . . . 4  |-  ( ( U  e.  V  /\  A  =  B )  ->  A  =  B )
108, 9nsyl3 111 . . 3  |-  ( ( U  e.  V  /\  A  =  B )  ->  -.  ( A  =/= 
B  /\  ( U  =  A  \/  U  =  B ) ) )
116, 102falsed 340 . 2  |-  ( ( U  e.  V  /\  A  =  B )  ->  ( U  e.  (/)  <->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B )
) ) )
12 elprg 3670 . . 3  |-  ( U  e.  V  ->  ( U  e.  { A ,  B }  <->  ( U  =  A  \/  U  =  B ) ) )
13 df-ne 2461 . . . 4  |-  ( A  =/=  B  <->  -.  A  =  B )
14 ibar 490 . . . 4  |-  ( A  =/=  B  ->  (
( U  =  A  \/  U  =  B )  <->  ( A  =/= 
B  /\  ( U  =  A  \/  U  =  B ) ) ) )
1513, 14sylbir 204 . . 3  |-  ( -.  A  =  B  -> 
( ( U  =  A  \/  U  =  B )  <->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B ) ) ) )
1612, 15sylan9bb 680 . 2  |-  ( ( U  e.  V  /\  -.  A  =  B
)  ->  ( U  e.  { A ,  B } 
<->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B ) ) ) )
172, 4, 11, 16ifbothda 3608 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 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468   ifcif 3578   {cpr 3654
This theorem is referenced by:  eupath2lem2  23917  eupath2lem3  23918
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660
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