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Theorem inttpemp 25139
Description: Two ways of saying that two triples have no common element. (Contributed by FL, 29-May-2014.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
inttpemp  |-  ( ( A  e.  G  /\  B  e.  H  /\  C  e.  I )  ->  ( ( { A ,  B ,  C }  i^i  { D ,  E ,  F } )  =  (/) 
<->  ( ( A  =/= 
D  /\  A  =/=  E  /\  A  =/=  F
)  /\  ( B  =/=  D  /\  B  =/= 
E  /\  B  =/=  F )  /\  ( C  =/=  D  /\  C  =/=  E  /\  C  =/= 
F ) ) ) )

Proof of Theorem inttpemp
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 disj 3495 . 2  |-  ( ( { A ,  B ,  C }  i^i  { D ,  E ,  F } )  =  (/)  <->  A. x  e.  { A ,  B ,  C }  -.  x  e.  { D ,  E ,  F }
)
2 eleq1 2343 . . . . . 6  |-  ( x  =  A  ->  (
x  e.  { D ,  E ,  F }  <->  A  e.  { D ,  E ,  F }
) )
3 id 19 . . . . . . . 8  |-  ( x  =  A  ->  x  =  A )
4 vex 2791 . . . . . . . 8  |-  x  e. 
_V
53, 4syl6eqelr 2372 . . . . . . 7  |-  ( x  =  A  ->  A  e.  _V )
6 eltpg 3676 . . . . . . 7  |-  ( A  e.  _V  ->  ( A  e.  { D ,  E ,  F }  <->  ( A  =  D  \/  A  =  E  \/  A  =  F )
) )
75, 6syl 15 . . . . . 6  |-  ( x  =  A  ->  ( A  e.  { D ,  E ,  F }  <->  ( A  =  D  \/  A  =  E  \/  A  =  F )
) )
82, 7bitrd 244 . . . . 5  |-  ( x  =  A  ->  (
x  e.  { D ,  E ,  F }  <->  ( A  =  D  \/  A  =  E  \/  A  =  F )
) )
98notbid 285 . . . 4  |-  ( x  =  A  ->  ( -.  x  e.  { D ,  E ,  F }  <->  -.  ( A  =  D  \/  A  =  E  \/  A  =  F ) ) )
10 ne3anior 2532 . . . 4  |-  ( ( A  =/=  D  /\  A  =/=  E  /\  A  =/=  F )  <->  -.  ( A  =  D  \/  A  =  E  \/  A  =  F )
)
119, 10syl6bbr 254 . . 3  |-  ( x  =  A  ->  ( -.  x  e.  { D ,  E ,  F }  <->  ( A  =/=  D  /\  A  =/=  E  /\  A  =/=  F ) ) )
12 eleq1 2343 . . . . . 6  |-  ( x  =  B  ->  (
x  e.  { D ,  E ,  F }  <->  B  e.  { D ,  E ,  F }
) )
13 id 19 . . . . . . . 8  |-  ( x  =  B  ->  x  =  B )
1413, 4syl6eqelr 2372 . . . . . . 7  |-  ( x  =  B  ->  B  e.  _V )
15 eltpg 3676 . . . . . . 7  |-  ( B  e.  _V  ->  ( B  e.  { D ,  E ,  F }  <->  ( B  =  D  \/  B  =  E  \/  B  =  F )
) )
1614, 15syl 15 . . . . . 6  |-  ( x  =  B  ->  ( B  e.  { D ,  E ,  F }  <->  ( B  =  D  \/  B  =  E  \/  B  =  F )
) )
1712, 16bitrd 244 . . . . 5  |-  ( x  =  B  ->  (
x  e.  { D ,  E ,  F }  <->  ( B  =  D  \/  B  =  E  \/  B  =  F )
) )
1817notbid 285 . . . 4  |-  ( x  =  B  ->  ( -.  x  e.  { D ,  E ,  F }  <->  -.  ( B  =  D  \/  B  =  E  \/  B  =  F ) ) )
19 ne3anior 2532 . . . 4  |-  ( ( B  =/=  D  /\  B  =/=  E  /\  B  =/=  F )  <->  -.  ( B  =  D  \/  B  =  E  \/  B  =  F )
)
2018, 19syl6bbr 254 . . 3  |-  ( x  =  B  ->  ( -.  x  e.  { D ,  E ,  F }  <->  ( B  =/=  D  /\  B  =/=  E  /\  B  =/=  F ) ) )
21 eleq1 2343 . . . . . 6  |-  ( x  =  C  ->  (
x  e.  { D ,  E ,  F }  <->  C  e.  { D ,  E ,  F }
) )
22 id 19 . . . . . . . 8  |-  ( x  =  C  ->  x  =  C )
2322, 4syl6eqelr 2372 . . . . . . 7  |-  ( x  =  C  ->  C  e.  _V )
24 eltpg 3676 . . . . . . 7  |-  ( C  e.  _V  ->  ( C  e.  { D ,  E ,  F }  <->  ( C  =  D  \/  C  =  E  \/  C  =  F )
) )
2523, 24syl 15 . . . . . 6  |-  ( x  =  C  ->  ( C  e.  { D ,  E ,  F }  <->  ( C  =  D  \/  C  =  E  \/  C  =  F )
) )
2621, 25bitrd 244 . . . . 5  |-  ( x  =  C  ->  (
x  e.  { D ,  E ,  F }  <->  ( C  =  D  \/  C  =  E  \/  C  =  F )
) )
2726notbid 285 . . . 4  |-  ( x  =  C  ->  ( -.  x  e.  { D ,  E ,  F }  <->  -.  ( C  =  D  \/  C  =  E  \/  C  =  F ) ) )
28 ne3anior 2532 . . . 4  |-  ( ( C  =/=  D  /\  C  =/=  E  /\  C  =/=  F )  <->  -.  ( C  =  D  \/  C  =  E  \/  C  =  F )
)
2927, 28syl6bbr 254 . . 3  |-  ( x  =  C  ->  ( -.  x  e.  { D ,  E ,  F }  <->  ( C  =/=  D  /\  C  =/=  E  /\  C  =/=  F ) ) )
3011, 20, 29raltpg 3684 . 2  |-  ( ( A  e.  G  /\  B  e.  H  /\  C  e.  I )  ->  ( A. x  e. 
{ A ,  B ,  C }  -.  x  e.  { D ,  E ,  F }  <->  ( ( A  =/=  D  /\  A  =/=  E  /\  A  =/= 
F )  /\  ( B  =/=  D  /\  B  =/=  E  /\  B  =/= 
F )  /\  ( C  =/=  D  /\  C  =/=  E  /\  C  =/= 
F ) ) ) )
311, 30syl5bb 248 1  |-  ( ( A  e.  G  /\  B  e.  H  /\  C  e.  I )  ->  ( ( { A ,  B ,  C }  i^i  { D ,  E ,  F } )  =  (/) 
<->  ( ( A  =/= 
D  /\  A  =/=  E  /\  A  =/=  F
)  /\  ( B  =/=  D  /\  B  =/= 
E  /\  B  =/=  F )  /\  ( C  =/=  D  /\  C  =/=  E  /\  C  =/= 
F ) ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ w3o 933    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   _Vcvv 2788    i^i cin 3151   (/)c0 3455   {ctp 3642
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-nul 3456  df-sn 3646  df-pr 3647  df-tp 3648
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