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Theorem otsndisj 4752
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 385 . . . . 5  |-  ( c  =  d  ->  (
c  =  d  \/  ( { <. A ,  B ,  c >. }  i^i  { <. A ,  B ,  d >. } )  =  (/) ) )
21a1d 25 . . . 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 4730 . . . . . . . . . . . . 13  |-  ( ( A  e.  X  /\  B  e.  Y  /\  c  e.  V )  ->  ( <. A ,  B ,  c >.  =  <. A ,  B ,  d
>. 
<->  ( A  =  A  /\  B  =  B  /\  c  =  d ) ) )
433expa 1196 . . . . . . . . . . . 12  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  c  e.  V )  ->  ( <. A ,  B , 
c >.  =  <. A ,  B ,  d >.  <->  ( A  =  A  /\  B  =  B  /\  c  =  d )
) )
54adantrr 716 . . . . . . . . . . 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 998 . . . . . . . . . . 11  |-  ( ( A  =  A  /\  B  =  B  /\  c  =  d )  ->  c  =  d )
75, 6syl6bi 228 . . . . . . . . . 10  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  ( c  e.  V  /\  d  e.  V ) )  -> 
( <. A ,  B ,  c >.  =  <. A ,  B ,  d
>.  ->  c  =  d ) )
87con3rr3 136 . . . . . . . . 9  |-  ( -.  c  =  d  -> 
( ( ( A  e.  X  /\  B  e.  Y )  /\  (
c  e.  V  /\  d  e.  V )
)  ->  -.  <. A ,  B ,  c >.  = 
<. A ,  B , 
d >. ) )
98imp 429 . . . . . . . 8  |-  ( ( -.  c  =  d  /\  ( ( A  e.  X  /\  B  e.  Y )  /\  (
c  e.  V  /\  d  e.  V )
) )  ->  -.  <. A ,  B , 
c >.  =  <. A ,  B ,  d >. )
109neqned 2670 . . . . . . 7  |-  ( ( -.  c  =  d  /\  ( ( A  e.  X  /\  B  e.  Y )  /\  (
c  e.  V  /\  d  e.  V )
) )  ->  <. A ,  B ,  c >.  =/= 
<. A ,  B , 
d >. )
11 disjsn2 4089 . . . . . . 7  |-  ( <. A ,  B , 
c >.  =/=  <. A ,  B ,  d >.  -> 
( { <. A ,  B ,  c >. }  i^i  { <. A ,  B ,  d >. } )  =  (/) )
1210, 11syl 16 . . . . . 6  |-  ( ( -.  c  =  d  /\  ( ( A  e.  X  /\  B  e.  Y )  /\  (
c  e.  V  /\  d  e.  V )
) )  ->  ( { <. A ,  B ,  c >. }  i^i  {
<. A ,  B , 
d >. } )  =  (/) )
1312olcd 393 . . . . 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 434 . . . 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 164 . . 3  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  ( c  e.  V  /\  d  e.  V ) )  -> 
( c  =  d  \/  ( { <. A ,  B ,  c
>. }  i^i  { <. A ,  B ,  d
>. } )  =  (/) ) )
1615ralrimivva 2885 . 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 4224 . . . 4  |-  ( c  =  d  ->  <. A ,  B ,  c >.  = 
<. A ,  B , 
d >. )
1817sneqd 4039 . . 3  |-  ( c  =  d  ->  { <. A ,  B ,  c
>. }  =  { <. A ,  B ,  d
>. } )
1918disjor 4431 . 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 212 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 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814    i^i cin 3475   (/)c0 3785   {csn 4027   <.cotp 4035  Disj wdisj 4417
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-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rmo 2822  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-ot 4036  df-disj 4418
This theorem is referenced by:  usgreghash2spotv  24743
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