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Theorem disjpr2 4090
Description: The intersection of distinct unordered pairs is disjoint. (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 4030 . . . 4  |-  { C ,  D }  =  ( { C }  u.  { D } )
21a1i 11 . . 3  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  ->  { C ,  D }  =  ( { C }  u.  { D } ) )
32ineq2d 3700 . 2  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  { C ,  D } )  =  ( { A ,  B }  i^i  ( { C }  u.  { D } ) ) )
4 indi 3744 . . 3  |-  ( { A ,  B }  i^i  ( { C }  u.  { D } ) )  =  ( ( { A ,  B }  i^i  { C }
)  u.  ( { A ,  B }  i^i  { D } ) )
5 df-pr 4030 . . . . . . . 8  |-  { A ,  B }  =  ( { A }  u.  { B } )
65ineq1i 3696 . . . . . . 7  |-  ( { A ,  B }  i^i  { C } )  =  ( ( { A }  u.  { B } )  i^i  { C } )
7 indir 3746 . . . . . . 7  |-  ( ( { A }  u.  { B } )  i^i 
{ C } )  =  ( ( { A }  i^i  { C } )  u.  ( { B }  i^i  { C } ) )
86, 7eqtri 2496 . . . . . 6  |-  ( { A ,  B }  i^i  { C } )  =  ( ( { A }  i^i  { C } )  u.  ( { B }  i^i  { C } ) )
9 disjsn2 4089 . . . . . . . . . 10  |-  ( A  =/=  C  ->  ( { A }  i^i  { C } )  =  (/) )
109adantr 465 . . . . . . . . 9  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A }  i^i  { C } )  =  (/) )
1110adantr 465 . . . . . . . 8  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A }  i^i  { C } )  =  (/) )
12 disjsn2 4089 . . . . . . . . . 10  |-  ( B  =/=  C  ->  ( { B }  i^i  { C } )  =  (/) )
1312adantl 466 . . . . . . . . 9  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { B }  i^i  { C } )  =  (/) )
1413adantr 465 . . . . . . . 8  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { B }  i^i  { C } )  =  (/) )
1511, 14jca 532 . . . . . . 7  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A }  i^i  { C }
)  =  (/)  /\  ( { B }  i^i  { C } )  =  (/) ) )
16 un00 3862 . . . . . . 7  |-  ( ( ( { A }  i^i  { C } )  =  (/)  /\  ( { B }  i^i  { C } )  =  (/) ) 
<->  ( ( { A }  i^i  { C }
)  u.  ( { B }  i^i  { C } ) )  =  (/) )
1715, 16sylib 196 . . . . . 6  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A }  i^i  { C }
)  u.  ( { B }  i^i  { C } ) )  =  (/) )
188, 17syl5eq 2520 . . . . 5  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  { C } )  =  (/) )
195ineq1i 3696 . . . . . . 7  |-  ( { A ,  B }  i^i  { D } )  =  ( ( { A }  u.  { B } )  i^i  { D } )
20 indir 3746 . . . . . . 7  |-  ( ( { A }  u.  { B } )  i^i 
{ D } )  =  ( ( { A }  i^i  { D } )  u.  ( { B }  i^i  { D } ) )
2119, 20eqtri 2496 . . . . . 6  |-  ( { A ,  B }  i^i  { D } )  =  ( ( { A }  i^i  { D } )  u.  ( { B }  i^i  { D } ) )
22 disjsn2 4089 . . . . . . . . . 10  |-  ( A  =/=  D  ->  ( { A }  i^i  { D } )  =  (/) )
2322adantr 465 . . . . . . . . 9  |-  ( ( A  =/=  D  /\  B  =/=  D )  -> 
( { A }  i^i  { D } )  =  (/) )
2423adantl 466 . . . . . . . 8  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A }  i^i  { D } )  =  (/) )
25 disjsn2 4089 . . . . . . . . . 10  |-  ( B  =/=  D  ->  ( { B }  i^i  { D } )  =  (/) )
2625adantl 466 . . . . . . . . 9  |-  ( ( A  =/=  D  /\  B  =/=  D )  -> 
( { B }  i^i  { D } )  =  (/) )
2726adantl 466 . . . . . . . 8  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { B }  i^i  { D } )  =  (/) )
2824, 27jca 532 . . . . . . 7  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A }  i^i  { D }
)  =  (/)  /\  ( { B }  i^i  { D } )  =  (/) ) )
29 un00 3862 . . . . . . 7  |-  ( ( ( { A }  i^i  { D } )  =  (/)  /\  ( { B }  i^i  { D } )  =  (/) ) 
<->  ( ( { A }  i^i  { D }
)  u.  ( { B }  i^i  { D } ) )  =  (/) )
3028, 29sylib 196 . . . . . 6  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A }  i^i  { D }
)  u.  ( { B }  i^i  { D } ) )  =  (/) )
3121, 30syl5eq 2520 . . . . 5  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  { D } )  =  (/) )
3218, 31uneq12d 3659 . . . 4  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A ,  B }  i^i  { C } )  u.  ( { A ,  B }  i^i  { D } ) )  =  ( (/)  u.  (/) ) )
33 un0 3810 . . . 4  |-  ( (/)  u.  (/) )  =  (/)
3432, 33syl6eq 2524 . . 3  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A ,  B }  i^i  { C } )  u.  ( { A ,  B }  i^i  { D } ) )  =  (/) )
354, 34syl5eq 2520 . 2  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  ( { C }  u.  { D } ) )  =  (/) )
363, 35eqtrd 2508 1  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  { C ,  D } )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    =/= wne 2662    u. cun 3474    i^i cin 3475   (/)c0 3785   {csn 4027   {cpr 4029
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-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-sn 4028  df-pr 4030
This theorem is referenced by:  constr3lem4  24351  constr3lem6  24353
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