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Theorem otsndisj 4706
Description: The singletons consisting of ordered triples which have distinct third components are disjunct. (Contributed by Alexander van der Vekens, 10-Mar-2018.)
Assertion
Ref Expression
otsndisj  |-  ( ( A  e.  X  /\  B  e.  Y )  -> Disj  c  e.  V  { <. A ,  B , 
c >. } )
Distinct variable groups:    A, c    B, c    V, c    X, c    Y, c

Proof of Theorem otsndisj
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 orc 387 . . . . 5  |-  ( c  =  d  ->  (
c  =  d  \/  ( { <. A ,  B ,  c >. }  i^i  { <. A ,  B ,  d >. } )  =  (/) ) )
21a1d 26 . . . 4  |-  ( c  =  d  ->  (
( ( A  e.  X  /\  B  e.  Y )  /\  (
c  e.  V  /\  d  e.  V )
)  ->  ( c  =  d  \/  ( { <. A ,  B ,  c >. }  i^i  {
<. A ,  B , 
d >. } )  =  (/) ) ) )
3 otthg 4685 . . . . . . . . . . . . 13  |-  ( ( A  e.  X  /\  B  e.  Y  /\  c  e.  V )  ->  ( <. A ,  B ,  c >.  =  <. A ,  B ,  d
>. 
<->  ( A  =  A  /\  B  =  B  /\  c  =  d ) ) )
433expa 1208 . . . . . . . . . . . 12  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  c  e.  V )  ->  ( <. A ,  B , 
c >.  =  <. A ,  B ,  d >.  <->  ( A  =  A  /\  B  =  B  /\  c  =  d )
) )
54adantrr 723 . . . . . . . . . . 11  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  ( c  e.  V  /\  d  e.  V ) )  -> 
( <. A ,  B ,  c >.  =  <. A ,  B ,  d
>. 
<->  ( A  =  A  /\  B  =  B  /\  c  =  d ) ) )
6 simp3 1010 . . . . . . . . . . 11  |-  ( ( A  =  A  /\  B  =  B  /\  c  =  d )  ->  c  =  d )
75, 6syl6bi 232 . . . . . . . . . 10  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  ( c  e.  V  /\  d  e.  V ) )  -> 
( <. A ,  B ,  c >.  =  <. A ,  B ,  d
>.  ->  c  =  d ) )
87con3rr3 142 . . . . . . . . 9  |-  ( -.  c  =  d  -> 
( ( ( A  e.  X  /\  B  e.  Y )  /\  (
c  e.  V  /\  d  e.  V )
)  ->  -.  <. A ,  B ,  c >.  = 
<. A ,  B , 
d >. ) )
98imp 431 . . . . . . . 8  |-  ( ( -.  c  =  d  /\  ( ( A  e.  X  /\  B  e.  Y )  /\  (
c  e.  V  /\  d  e.  V )
) )  ->  -.  <. A ,  B , 
c >.  =  <. A ,  B ,  d >. )
109neqned 2631 . . . . . . 7  |-  ( ( -.  c  =  d  /\  ( ( A  e.  X  /\  B  e.  Y )  /\  (
c  e.  V  /\  d  e.  V )
) )  ->  <. A ,  B ,  c >.  =/= 
<. A ,  B , 
d >. )
11 disjsn2 4033 . . . . . . 7  |-  ( <. A ,  B , 
c >.  =/=  <. A ,  B ,  d >.  -> 
( { <. A ,  B ,  c >. }  i^i  { <. A ,  B ,  d >. } )  =  (/) )
1210, 11syl 17 . . . . . 6  |-  ( ( -.  c  =  d  /\  ( ( A  e.  X  /\  B  e.  Y )  /\  (
c  e.  V  /\  d  e.  V )
) )  ->  ( { <. A ,  B ,  c >. }  i^i  {
<. A ,  B , 
d >. } )  =  (/) )
1312olcd 395 . . . . 5  |-  ( ( -.  c  =  d  /\  ( ( A  e.  X  /\  B  e.  Y )  /\  (
c  e.  V  /\  d  e.  V )
) )  ->  (
c  =  d  \/  ( { <. A ,  B ,  c >. }  i^i  { <. A ,  B ,  d >. } )  =  (/) ) )
1413ex 436 . . . 4  |-  ( -.  c  =  d  -> 
( ( ( A  e.  X  /\  B  e.  Y )  /\  (
c  e.  V  /\  d  e.  V )
)  ->  ( c  =  d  \/  ( { <. A ,  B ,  c >. }  i^i  {
<. A ,  B , 
d >. } )  =  (/) ) ) )
152, 14pm2.61i 168 . . 3  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  ( c  e.  V  /\  d  e.  V ) )  -> 
( c  =  d  \/  ( { <. A ,  B ,  c
>. }  i^i  { <. A ,  B ,  d
>. } )  =  (/) ) )
1615ralrimivva 2809 . 2  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  A. c  e.  V  A. d  e.  V  ( c  =  d  \/  ( { <. A ,  B ,  c
>. }  i^i  { <. A ,  B ,  d
>. } )  =  (/) ) )
17 oteq3 4177 . . . 4  |-  ( c  =  d  ->  <. A ,  B ,  c >.  = 
<. A ,  B , 
d >. )
1817sneqd 3980 . . 3  |-  ( c  =  d  ->  { <. A ,  B ,  c
>. }  =  { <. A ,  B ,  d
>. } )
1918disjor 4387 . 2  |-  (Disj  c  e.  V  { <. A ,  B ,  c >. }  <->  A. c  e.  V  A. d  e.  V  ( c  =  d  \/  ( { <. A ,  B ,  c
>. }  i^i  { <. A ,  B ,  d
>. } )  =  (/) ) )
2016, 19sylibr 216 1  |-  ( ( A  e.  X  /\  B  e.  Y )  -> Disj  c  e.  V  { <. A ,  B , 
c >. } )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887    =/= wne 2622   A.wral 2737    i^i cin 3403   (/)c0 3731   {csn 3968   <.cotp 3976  Disj wdisj 4373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rmo 2745  df-rab 2746  df-v 3047  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-ot 3977  df-disj 4374
This theorem is referenced by:  usgreghash2spotv  25794
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