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Theorem eupath2lem1 23901
Description: Lemma for eupath2 23904. (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 2344 . . 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 2344 . . 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 3459 . . . 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 2462 . . . 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 3657 . . 3  |-  ( U  e.  V  ->  ( U  e.  { A ,  B }  <->  ( U  =  A  \/  U  =  B ) ) )
13 df-ne 2448 . . . 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 3595 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 1623    e. wcel 1684    =/= wne 2446   (/)c0 3455   ifcif 3565   {cpr 3641
This theorem is referenced by:  eupath2lem2  23902  eupath2lem3  23903
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-un 3157  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647
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