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

Theorem otsndisj 4739
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 4717 . . . . . . . . . . . . 13  |-  ( ( A  e.  X  /\  B  e.  Y  /\  c  e.  V )  ->  ( <. A ,  B ,  c >.  =  <. A ,  B ,  d
>. 
<->  ( A  =  A  /\  B  =  B  /\  c  =  d ) ) )
433expa 1195 . . . . . . . . . . . 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 997 . . . . . . . . . . 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 2644 . . . . . . 7  |-  ( ( -.  c  =  d  /\  ( ( A  e.  X  /\  B  e.  Y )  /\  (
c  e.  V  /\  d  e.  V )
) )  ->  <. A ,  B ,  c >.  =/= 
<. A ,  B , 
d >. )
11 disjsn2 4073 . . . . . . 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 2862 . 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 4210 . . . 4  |-  ( c  =  d  ->  <. A ,  B ,  c >.  = 
<. A ,  B , 
d >. )
1817sneqd 4023 . . 3  |-  ( c  =  d  ->  { <. A ,  B ,  c
>. }  =  { <. A ,  B ,  d
>. } )
1918disjor 4418 . 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 972    = wceq 1381    e. wcel 1802    =/= wne 2636   A.wral 2791    i^i cin 3458   (/)c0 3768   {csn 4011   <.cotp 4019  Disj wdisj 4404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4555  ax-nul 4563  ax-pr 4673
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rmo 2799  df-rab 2800  df-v 3095  df-dif 3462  df-un 3464  df-in 3466  df-ss 3473  df-nul 3769  df-if 3924  df-sn 4012  df-pr 4014  df-op 4018  df-ot 4020  df-disj 4405
This theorem is referenced by:  usgreghash2spotv  24935
  Copyright terms: Public domain W3C validator