MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eupath2lem1 Structured version   Unicode version

Theorem eupath2lem1 24750
Description: Lemma for eupath2 24753. (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 2540 . . 3  |-  ( (/)  =  if ( A  =  B ,  (/) ,  { A ,  B }
)  ->  ( U  e.  (/)  <->  U  e.  if ( A  =  B ,  (/) ,  { A ,  B } ) ) )
21bibi1d 319 . 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 2540 . . 3  |-  ( { A ,  B }  =  if ( A  =  B ,  (/) ,  { A ,  B }
)  ->  ( U  e.  { A ,  B } 
<->  U  e.  if ( A  =  B ,  (/)
,  { A ,  B } ) ) )
43bibi1d 319 . 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 3789 . . . 4  |-  -.  U  e.  (/)
65a1i 11 . . 3  |-  ( ( U  e.  V  /\  A  =  B )  ->  -.  U  e.  (/) )
7 simpl 457 . . . . 5  |-  ( ( A  =/=  B  /\  ( U  =  A  \/  U  =  B
) )  ->  A  =/=  B )
87neneqd 2669 . . . 4  |-  ( ( A  =/=  B  /\  ( U  =  A  \/  U  =  B
) )  ->  -.  A  =  B )
9 simpr 461 . . . 4  |-  ( ( U  e.  V  /\  A  =  B )  ->  A  =  B )
108, 9nsyl3 119 . . 3  |-  ( ( U  e.  V  /\  A  =  B )  ->  -.  ( A  =/= 
B  /\  ( U  =  A  \/  U  =  B ) ) )
116, 102falsed 351 . 2  |-  ( ( U  e.  V  /\  A  =  B )  ->  ( U  e.  (/)  <->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B )
) ) )
12 elprg 4043 . . 3  |-  ( U  e.  V  ->  ( U  e.  { A ,  B }  <->  ( U  =  A  \/  U  =  B ) ) )
13 df-ne 2664 . . . 4  |-  ( A  =/=  B  <->  -.  A  =  B )
14 ibar 504 . . . 4  |-  ( A  =/=  B  ->  (
( U  =  A  \/  U  =  B )  <->  ( A  =/= 
B  /\  ( U  =  A  \/  U  =  B ) ) ) )
1513, 14sylbir 213 . . 3  |-  ( -.  A  =  B  -> 
( ( U  =  A  \/  U  =  B )  <->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B ) ) ) )
1612, 15sylan9bb 699 . 2  |-  ( ( U  e.  V  /\  -.  A  =  B
)  ->  ( U  e.  { A ,  B } 
<->  ( A  =/=  B  /\  ( U  =  A  \/  U  =  B ) ) ) )
172, 4, 11, 16ifbothda 3974 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 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   (/)c0 3785   ifcif 3939   {cpr 4029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-dif 3479  df-un 3481  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030
This theorem is referenced by:  eupath2lem2  24751  eupath2lem3  24752
  Copyright terms: Public domain W3C validator