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Theorem disjpr2 27409
Description: The intersection of distinct pairs is disjoint, analogous to disjsn2 3770. (Contributed by Alexander van der Vekens, 11-Nov-2017.)
Assertion
Ref Expression
disjpr2  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  { C ,  D } )  =  (/) )

Proof of Theorem disjpr2
StepHypRef Expression
1 df-pr 3723 . . . 4  |-  { C ,  D }  =  ( { C }  u.  { D } )
21a1i 10 . . 3  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  ->  { C ,  D }  =  ( { C }  u.  { D } ) )
32ineq2d 3446 . 2  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  { C ,  D } )  =  ( { A ,  B }  i^i  ( { C }  u.  { D } ) ) )
4 indi 3491 . . 3  |-  ( { A ,  B }  i^i  ( { C }  u.  { D } ) )  =  ( ( { A ,  B }  i^i  { C }
)  u.  ( { A ,  B }  i^i  { D } ) )
5 df-pr 3723 . . . . . . . 8  |-  { A ,  B }  =  ( { A }  u.  { B } )
65ineq1i 3442 . . . . . . 7  |-  ( { A ,  B }  i^i  { C } )  =  ( ( { A }  u.  { B } )  i^i  { C } )
7 indir 3493 . . . . . . 7  |-  ( ( { A }  u.  { B } )  i^i 
{ C } )  =  ( ( { A }  i^i  { C } )  u.  ( { B }  i^i  { C } ) )
86, 7eqtri 2378 . . . . . 6  |-  ( { A ,  B }  i^i  { C } )  =  ( ( { A }  i^i  { C } )  u.  ( { B }  i^i  { C } ) )
9 disjsn2 3770 . . . . . . . . . 10  |-  ( A  =/=  C  ->  ( { A }  i^i  { C } )  =  (/) )
109adantr 451 . . . . . . . . 9  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A }  i^i  { C } )  =  (/) )
1110adantr 451 . . . . . . . 8  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A }  i^i  { C } )  =  (/) )
12 disjsn2 3770 . . . . . . . . . 10  |-  ( B  =/=  C  ->  ( { B }  i^i  { C } )  =  (/) )
1312adantl 452 . . . . . . . . 9  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { B }  i^i  { C } )  =  (/) )
1413adantr 451 . . . . . . . 8  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { B }  i^i  { C } )  =  (/) )
1511, 14jca 518 . . . . . . 7  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A }  i^i  { C }
)  =  (/)  /\  ( { B }  i^i  { C } )  =  (/) ) )
16 un00 3566 . . . . . . 7  |-  ( ( ( { A }  i^i  { C } )  =  (/)  /\  ( { B }  i^i  { C } )  =  (/) ) 
<->  ( ( { A }  i^i  { C }
)  u.  ( { B }  i^i  { C } ) )  =  (/) )
1715, 16sylib 188 . . . . . 6  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A }  i^i  { C }
)  u.  ( { B }  i^i  { C } ) )  =  (/) )
188, 17syl5eq 2402 . . . . 5  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  { C } )  =  (/) )
195ineq1i 3442 . . . . . . 7  |-  ( { A ,  B }  i^i  { D } )  =  ( ( { A }  u.  { B } )  i^i  { D } )
20 indir 3493 . . . . . . 7  |-  ( ( { A }  u.  { B } )  i^i 
{ D } )  =  ( ( { A }  i^i  { D } )  u.  ( { B }  i^i  { D } ) )
2119, 20eqtri 2378 . . . . . 6  |-  ( { A ,  B }  i^i  { D } )  =  ( ( { A }  i^i  { D } )  u.  ( { B }  i^i  { D } ) )
22 disjsn2 3770 . . . . . . . . . 10  |-  ( A  =/=  D  ->  ( { A }  i^i  { D } )  =  (/) )
2322adantr 451 . . . . . . . . 9  |-  ( ( A  =/=  D  /\  B  =/=  D )  -> 
( { A }  i^i  { D } )  =  (/) )
2423adantl 452 . . . . . . . 8  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A }  i^i  { D } )  =  (/) )
25 disjsn2 3770 . . . . . . . . . 10  |-  ( B  =/=  D  ->  ( { B }  i^i  { D } )  =  (/) )
2625adantl 452 . . . . . . . . 9  |-  ( ( A  =/=  D  /\  B  =/=  D )  -> 
( { B }  i^i  { D } )  =  (/) )
2726adantl 452 . . . . . . . 8  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { B }  i^i  { D } )  =  (/) )
2824, 27jca 518 . . . . . . 7  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A }  i^i  { D }
)  =  (/)  /\  ( { B }  i^i  { D } )  =  (/) ) )
29 un00 3566 . . . . . . 7  |-  ( ( ( { A }  i^i  { D } )  =  (/)  /\  ( { B }  i^i  { D } )  =  (/) ) 
<->  ( ( { A }  i^i  { D }
)  u.  ( { B }  i^i  { D } ) )  =  (/) )
3028, 29sylib 188 . . . . . 6  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A }  i^i  { D }
)  u.  ( { B }  i^i  { D } ) )  =  (/) )
3121, 30syl5eq 2402 . . . . 5  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  { D } )  =  (/) )
3218, 31uneq12d 3406 . . . 4  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A ,  B }  i^i  { C } )  u.  ( { A ,  B }  i^i  { D } ) )  =  ( (/)  u.  (/) ) )
33 un0 3555 . . . 4  |-  ( (/)  u.  (/) )  =  (/)
3432, 33syl6eq 2406 . . 3  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A ,  B }  i^i  { C } )  u.  ( { A ,  B }  i^i  { D } ) )  =  (/) )
354, 34syl5eq 2402 . 2  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  ( { C }  u.  { D } ) )  =  (/) )
363, 35eqtrd 2390 1  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  { C ,  D } )  =  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1642    =/= wne 2521    u. cun 3226    i^i cin 3227   (/)c0 3531   {csn 3716   {cpr 3717
This theorem is referenced by:  constr3lem4  27754  constr3lem6  27756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-sn 3722  df-pr 3723
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