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Theorem exmidapne 7259
Description: Excluded middle implies there is only one tight apartness on any class, namely negated equality. (Contributed by Jim Kingdon, 14-Feb-2025.)
Assertion
Ref Expression
exmidapne  |-  (EXMID  ->  ( R TAp  A  <->  R  =  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A )  /\  u  =/=  v ) } ) )
Distinct variable group:    u, A, v
Allowed substitution hints:    R( v, u)

Proof of Theorem exmidapne
Dummy variables  p  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 528 . . . . . . 7  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  ->  R TAp  A )
2 simpr 110 . . . . . . 7  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  ->  p  e.  R )
3 dftap2 7250 . . . . . . . . . 10  |-  ( R TAp 
A  <->  ( R  C_  ( A  X.  A
)  /\  ( A. x  e.  A  -.  x R x  /\  A. x  e.  A  A. y  e.  A  (
x R y  -> 
y R x ) )  /\  ( A. x  e.  A  A. y  e.  A  A. z  e.  A  (
x R y  -> 
( x R z  \/  y R z ) )  /\  A. x  e.  A  A. y  e.  A  ( -.  x R y  ->  x  =  y )
) ) )
43biimpi 120 . . . . . . . . 9  |-  ( R TAp 
A  ->  ( R  C_  ( A  X.  A
)  /\  ( A. x  e.  A  -.  x R x  /\  A. x  e.  A  A. y  e.  A  (
x R y  -> 
y R x ) )  /\  ( A. x  e.  A  A. y  e.  A  A. z  e.  A  (
x R y  -> 
( x R z  \/  y R z ) )  /\  A. x  e.  A  A. y  e.  A  ( -.  x R y  ->  x  =  y )
) ) )
54simp1d 1009 . . . . . . . 8  |-  ( R TAp 
A  ->  R  C_  ( A  X.  A ) )
65sseld 3155 . . . . . . 7  |-  ( R TAp 
A  ->  ( p  e.  R  ->  p  e.  ( A  X.  A
) ) )
71, 2, 6sylc 62 . . . . . 6  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  ->  p  e.  ( A  X.  A
) )
8 1st2nd2 6176 . . . . . 6  |-  ( p  e.  ( A  X.  A )  ->  p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
97, 8syl 14 . . . . 5  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  ->  p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
10 xp1st 6166 . . . . . . . 8  |-  ( p  e.  ( A  X.  A )  ->  ( 1st `  p )  e.  A )
117, 10syl 14 . . . . . . 7  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  ->  ( 1st `  p )  e.  A
)
12 xp2nd 6167 . . . . . . . 8  |-  ( p  e.  ( A  X.  A )  ->  ( 2nd `  p )  e.  A )
137, 12syl 14 . . . . . . 7  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  ->  ( 2nd `  p )  e.  A
)
149, 2eqeltrrd 2255 . . . . . . . . . 10  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  ->  <. ( 1st `  p ) ,  ( 2nd `  p )
>.  e.  R )
1514adantr 276 . . . . . . . . 9  |-  ( ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  /\  ( 1st `  p )  =  ( 2nd `  p ) )  ->  <. ( 1st `  p ) ,  ( 2nd `  p )
>.  e.  R )
16 simpr 110 . . . . . . . . . . 11  |-  ( ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  /\  ( 1st `  p )  =  ( 2nd `  p ) )  ->  ( 1st `  p )  =  ( 2nd `  p ) )
1716opeq2d 3786 . . . . . . . . . 10  |-  ( ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  /\  ( 1st `  p )  =  ( 2nd `  p ) )  ->  <. ( 1st `  p ) ,  ( 1st `  p )
>.  =  <. ( 1st `  p ) ,  ( 2nd `  p )
>. )
18 id 19 . . . . . . . . . . . . . 14  |-  ( x  =  ( 1st `  p
)  ->  x  =  ( 1st `  p ) )
1918, 18breq12d 4017 . . . . . . . . . . . . 13  |-  ( x  =  ( 1st `  p
)  ->  ( x R x  <->  ( 1st `  p
) R ( 1st `  p ) ) )
2019notbid 667 . . . . . . . . . . . 12  |-  ( x  =  ( 1st `  p
)  ->  ( -.  x R x  <->  -.  ( 1st `  p ) R ( 1st `  p
) ) )
214simp2d 1010 . . . . . . . . . . . . . 14  |-  ( R TAp 
A  ->  ( A. x  e.  A  -.  x R x  /\  A. x  e.  A  A. y  e.  A  (
x R y  -> 
y R x ) ) )
2221simpld 112 . . . . . . . . . . . . 13  |-  ( R TAp 
A  ->  A. x  e.  A  -.  x R x )
2322ad3antlr 493 . . . . . . . . . . . 12  |-  ( ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  /\  ( 1st `  p )  =  ( 2nd `  p ) )  ->  A. x  e.  A  -.  x R x )
2411adantr 276 . . . . . . . . . . . 12  |-  ( ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  /\  ( 1st `  p )  =  ( 2nd `  p ) )  ->  ( 1st `  p )  e.  A
)
2520, 23, 24rspcdva 2847 . . . . . . . . . . 11  |-  ( ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  /\  ( 1st `  p )  =  ( 2nd `  p ) )  ->  -.  ( 1st `  p ) R ( 1st `  p
) )
26 df-br 4005 . . . . . . . . . . 11  |-  ( ( 1st `  p ) R ( 1st `  p
)  <->  <. ( 1st `  p
) ,  ( 1st `  p ) >.  e.  R
)
2725, 26sylnib 676 . . . . . . . . . 10  |-  ( ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  /\  ( 1st `  p )  =  ( 2nd `  p ) )  ->  -.  <. ( 1st `  p ) ,  ( 1st `  p
) >.  e.  R )
2817, 27eqneltrrd 2274 . . . . . . . . 9  |-  ( ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  /\  ( 1st `  p )  =  ( 2nd `  p ) )  ->  -.  <. ( 1st `  p ) ,  ( 2nd `  p
) >.  e.  R )
2915, 28pm2.65da 661 . . . . . . . 8  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  ->  -.  ( 1st `  p )  =  ( 2nd `  p
) )
3029neqned 2354 . . . . . . 7  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  ->  ( 1st `  p )  =/=  ( 2nd `  p ) )
3111, 13, 30jca31 309 . . . . . 6  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  ->  ( (
( 1st `  p
)  e.  A  /\  ( 2nd `  p )  e.  A )  /\  ( 1st `  p )  =/=  ( 2nd `  p
) ) )
32 eleq1 2240 . . . . . . . . . 10  |-  ( u  =  ( 1st `  p
)  ->  ( u  e.  A  <->  ( 1st `  p
)  e.  A ) )
33 eleq1 2240 . . . . . . . . . 10  |-  ( v  =  ( 2nd `  p
)  ->  ( v  e.  A  <->  ( 2nd `  p
)  e.  A ) )
3432, 33bi2anan9 606 . . . . . . . . 9  |-  ( ( u  =  ( 1st `  p )  /\  v  =  ( 2nd `  p
) )  ->  (
( u  e.  A  /\  v  e.  A
)  <->  ( ( 1st `  p )  e.  A  /\  ( 2nd `  p
)  e.  A ) ) )
35 simpl 109 . . . . . . . . . 10  |-  ( ( u  =  ( 1st `  p )  /\  v  =  ( 2nd `  p
) )  ->  u  =  ( 1st `  p
) )
36 simpr 110 . . . . . . . . . 10  |-  ( ( u  =  ( 1st `  p )  /\  v  =  ( 2nd `  p
) )  ->  v  =  ( 2nd `  p
) )
3735, 36neeq12d 2367 . . . . . . . . 9  |-  ( ( u  =  ( 1st `  p )  /\  v  =  ( 2nd `  p
) )  ->  (
u  =/=  v  <->  ( 1st `  p )  =/=  ( 2nd `  p ) ) )
3834, 37anbi12d 473 . . . . . . . 8  |-  ( ( u  =  ( 1st `  p )  /\  v  =  ( 2nd `  p
) )  ->  (
( ( u  e.  A  /\  v  e.  A )  /\  u  =/=  v )  <->  ( (
( 1st `  p
)  e.  A  /\  ( 2nd `  p )  e.  A )  /\  ( 1st `  p )  =/=  ( 2nd `  p
) ) ) )
3938opelopabga 4264 . . . . . . 7  |-  ( ( ( 1st `  p
)  e.  A  /\  ( 2nd `  p )  e.  A )  -> 
( <. ( 1st `  p
) ,  ( 2nd `  p ) >.  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A )  /\  u  =/=  v ) }  <->  ( (
( 1st `  p
)  e.  A  /\  ( 2nd `  p )  e.  A )  /\  ( 1st `  p )  =/=  ( 2nd `  p
) ) ) )
4011, 13, 39syl2anc 411 . . . . . 6  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  ->  ( <. ( 1st `  p ) ,  ( 2nd `  p
) >.  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) }  <->  ( (
( 1st `  p
)  e.  A  /\  ( 2nd `  p )  e.  A )  /\  ( 1st `  p )  =/=  ( 2nd `  p
) ) ) )
4131, 40mpbird 167 . . . . 5  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  ->  <. ( 1st `  p ) ,  ( 2nd `  p )
>.  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )
429, 41eqeltrd 2254 . . . 4  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  R
)  ->  p  e.  {
<. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A )  /\  u  =/=  v ) } )
43 relopab 4754 . . . . . . 7  |-  Rel  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A )  /\  u  =/=  v ) }
44 1st2nd 6182 . . . . . . 7  |-  ( ( Rel  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) }  /\  p  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )  ->  p  =  <. ( 1st `  p ) ,  ( 2nd `  p
) >. )
4543, 44mpan 424 . . . . . 6  |-  ( p  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) }  ->  p  =  <. ( 1st `  p
) ,  ( 2nd `  p ) >. )
4645adantl 277 . . . . 5  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )  ->  p  =  <. ( 1st `  p ) ,  ( 2nd `  p
) >. )
47 breq2 4008 . . . . . . . . . 10  |-  ( y  =  ( 2nd `  p
)  ->  ( ( 1st `  p ) R y  <->  ( 1st `  p
) R ( 2nd `  p ) ) )
4847notbid 667 . . . . . . . . 9  |-  ( y  =  ( 2nd `  p
)  ->  ( -.  ( 1st `  p ) R y  <->  -.  ( 1st `  p ) R ( 2nd `  p
) ) )
49 eqeq2 2187 . . . . . . . . 9  |-  ( y  =  ( 2nd `  p
)  ->  ( ( 1st `  p )  =  y  <->  ( 1st `  p
)  =  ( 2nd `  p ) ) )
5048, 49imbi12d 234 . . . . . . . 8  |-  ( y  =  ( 2nd `  p
)  ->  ( ( -.  ( 1st `  p
) R y  -> 
( 1st `  p
)  =  y )  <-> 
( -.  ( 1st `  p ) R ( 2nd `  p )  ->  ( 1st `  p
)  =  ( 2nd `  p ) ) ) )
51 breq1 4007 . . . . . . . . . . . 12  |-  ( x  =  ( 1st `  p
)  ->  ( x R y  <->  ( 1st `  p ) R y ) )
5251notbid 667 . . . . . . . . . . 11  |-  ( x  =  ( 1st `  p
)  ->  ( -.  x R y  <->  -.  ( 1st `  p ) R y ) )
53 eqeq1 2184 . . . . . . . . . . 11  |-  ( x  =  ( 1st `  p
)  ->  ( x  =  y  <->  ( 1st `  p
)  =  y ) )
5452, 53imbi12d 234 . . . . . . . . . 10  |-  ( x  =  ( 1st `  p
)  ->  ( ( -.  x R y  ->  x  =  y )  <->  ( -.  ( 1st `  p
) R y  -> 
( 1st `  p
)  =  y ) ) )
5554ralbidv 2477 . . . . . . . . 9  |-  ( x  =  ( 1st `  p
)  ->  ( A. y  e.  A  ( -.  x R y  ->  x  =  y )  <->  A. y  e.  A  ( -.  ( 1st `  p
) R y  -> 
( 1st `  p
)  =  y ) ) )
564simp3d 1011 . . . . . . . . . . 11  |-  ( R TAp 
A  ->  ( A. x  e.  A  A. y  e.  A  A. z  e.  A  (
x R y  -> 
( x R z  \/  y R z ) )  /\  A. x  e.  A  A. y  e.  A  ( -.  x R y  ->  x  =  y )
) )
5756simprd 114 . . . . . . . . . 10  |-  ( R TAp 
A  ->  A. x  e.  A  A. y  e.  A  ( -.  x R y  ->  x  =  y ) )
5857ad2antlr 489 . . . . . . . . 9  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )  ->  A. x  e.  A  A. y  e.  A  ( -.  x R
y  ->  x  =  y ) )
5932anbi1d 465 . . . . . . . . . . . . . 14  |-  ( u  =  ( 1st `  p
)  ->  ( (
u  e.  A  /\  v  e.  A )  <->  ( ( 1st `  p
)  e.  A  /\  v  e.  A )
) )
60 neeq1 2360 . . . . . . . . . . . . . 14  |-  ( u  =  ( 1st `  p
)  ->  ( u  =/=  v  <->  ( 1st `  p
)  =/=  v ) )
6159, 60anbi12d 473 . . . . . . . . . . . . 13  |-  ( u  =  ( 1st `  p
)  ->  ( (
( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v )  <->  ( (
( 1st `  p
)  e.  A  /\  v  e.  A )  /\  ( 1st `  p
)  =/=  v ) ) )
6233anbi2d 464 . . . . . . . . . . . . . 14  |-  ( v  =  ( 2nd `  p
)  ->  ( (
( 1st `  p
)  e.  A  /\  v  e.  A )  <->  ( ( 1st `  p
)  e.  A  /\  ( 2nd `  p )  e.  A ) ) )
63 neeq2 2361 . . . . . . . . . . . . . 14  |-  ( v  =  ( 2nd `  p
)  ->  ( ( 1st `  p )  =/=  v  <->  ( 1st `  p
)  =/=  ( 2nd `  p ) ) )
6462, 63anbi12d 473 . . . . . . . . . . . . 13  |-  ( v  =  ( 2nd `  p
)  ->  ( (
( ( 1st `  p
)  e.  A  /\  v  e.  A )  /\  ( 1st `  p
)  =/=  v )  <-> 
( ( ( 1st `  p )  e.  A  /\  ( 2nd `  p
)  e.  A )  /\  ( 1st `  p
)  =/=  ( 2nd `  p ) ) ) )
6561, 64elopabi 6196 . . . . . . . . . . . 12  |-  ( p  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) }  ->  ( ( ( 1st `  p
)  e.  A  /\  ( 2nd `  p )  e.  A )  /\  ( 1st `  p )  =/=  ( 2nd `  p
) ) )
6665adantl 277 . . . . . . . . . . 11  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )  ->  ( ( ( 1st `  p )  e.  A  /\  ( 2nd `  p )  e.  A )  /\  ( 1st `  p )  =/=  ( 2nd `  p
) ) )
6766simpld 112 . . . . . . . . . 10  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )  ->  ( ( 1st `  p )  e.  A  /\  ( 2nd `  p
)  e.  A ) )
6867simpld 112 . . . . . . . . 9  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )  ->  ( 1st `  p
)  e.  A )
6955, 58, 68rspcdva 2847 . . . . . . . 8  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )  ->  A. y  e.  A  ( -.  ( 1st `  p ) R y  ->  ( 1st `  p
)  =  y ) )
7067simprd 114 . . . . . . . 8  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )  ->  ( 2nd `  p
)  e.  A )
7150, 69, 70rspcdva 2847 . . . . . . 7  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )  ->  ( -.  ( 1st `  p ) R ( 2nd `  p
)  ->  ( 1st `  p )  =  ( 2nd `  p ) ) )
7266simprd 114 . . . . . . . 8  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )  ->  ( 1st `  p
)  =/=  ( 2nd `  p ) )
7372neneqd 2368 . . . . . . 7  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )  ->  -.  ( 1st `  p )  =  ( 2nd `  p ) )
74 exmidexmid 4197 . . . . . . . . 9  |-  (EXMID  -> DECID  ( 1st `  p
) R ( 2nd `  p ) )
75 con1dc 856 . . . . . . . . 9  |-  (DECID  ( 1st `  p ) R ( 2nd `  p )  ->  ( ( -.  ( 1st `  p
) R ( 2nd `  p )  ->  ( 1st `  p )  =  ( 2nd `  p
) )  ->  ( -.  ( 1st `  p
)  =  ( 2nd `  p )  ->  ( 1st `  p ) R ( 2nd `  p
) ) ) )
7674, 75syl 14 . . . . . . . 8  |-  (EXMID  ->  (
( -.  ( 1st `  p ) R ( 2nd `  p )  ->  ( 1st `  p
)  =  ( 2nd `  p ) )  -> 
( -.  ( 1st `  p )  =  ( 2nd `  p )  ->  ( 1st `  p
) R ( 2nd `  p ) ) ) )
7776ad2antrr 488 . . . . . . 7  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )  ->  ( ( -.  ( 1st `  p
) R ( 2nd `  p )  ->  ( 1st `  p )  =  ( 2nd `  p
) )  ->  ( -.  ( 1st `  p
)  =  ( 2nd `  p )  ->  ( 1st `  p ) R ( 2nd `  p
) ) ) )
7871, 73, 77mp2d 47 . . . . . 6  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )  ->  ( 1st `  p
) R ( 2nd `  p ) )
79 df-br 4005 . . . . . 6  |-  ( ( 1st `  p ) R ( 2nd `  p
)  <->  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  e.  R
)
8078, 79sylib 122 . . . . 5  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )  ->  <. ( 1st `  p
) ,  ( 2nd `  p ) >.  e.  R
)
8146, 80eqeltrd 2254 . . . 4  |-  ( ( (EXMID 
/\  R TAp  A )  /\  p  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )  ->  p  e.  R
)
8242, 81impbida 596 . . 3  |-  ( (EXMID  /\  R TAp  A )  -> 
( p  e.  R  <->  p  e.  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } ) )
8382eqrdv 2175 . 2  |-  ( (EXMID  /\  R TAp  A )  ->  R  =  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )
84 exmidexmid 4197 . . . . . . 7  |-  (EXMID  -> DECID  x  =  y
)
8584ralrimivw 2551 . . . . . 6  |-  (EXMID  ->  A. y  e.  A DECID  x  =  y
)
8685ralrimivw 2551 . . . . 5  |-  (EXMID  ->  A. x  e.  A  A. y  e.  A DECID  x  =  y
)
87 netap 7253 . . . . 5  |-  ( A. x  e.  A  A. y  e.  A DECID  x  =  y  ->  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } TAp  A
)
8886, 87syl 14 . . . 4  |-  (EXMID  ->  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } TAp  A
)
8988adantr 276 . . 3  |-  ( (EXMID  /\  R  =  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )  ->  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } TAp  A
)
90 tapeq1 7251 . . . 4  |-  ( R  =  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) }  ->  ( R TAp  A  <->  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } TAp  A
) )
9190adantl 277 . . 3  |-  ( (EXMID  /\  R  =  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )  ->  ( R TAp  A  <->  {
<. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A )  /\  u  =/=  v ) } TAp  A
) )
9289, 91mpbird 167 . 2  |-  ( (EXMID  /\  R  =  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A
)  /\  u  =/=  v ) } )  ->  R TAp  A )
9383, 92impbida 596 1  |-  (EXMID  ->  ( R TAp  A  <->  R  =  { <. u ,  v >.  |  ( ( u  e.  A  /\  v  e.  A )  /\  u  =/=  v ) } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708  DECID wdc 834    /\ w3a 978    = wceq 1353    e. wcel 2148    =/= wne 2347   A.wral 2455    C_ wss 3130   <.cop 3596   class class class wbr 4004   {copab 4064  EXMIDwem 4195    X. cxp 4625   Rel wrel 4632   ` cfv 5217   1stc1st 6139   2ndc2nd 6140   TAp wtap 7248
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-exmid 4196  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fo 5223  df-fv 5225  df-1st 6141  df-2nd 6142  df-pap 7247  df-tap 7249
This theorem is referenced by:  exmidmotap  7260
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