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Theorem disjpr2 3501
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 3448 . . . 4  |-  { C ,  D }  =  ( { C }  u.  { D } )
21a1i 9 . . 3  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  ->  { C ,  D }  =  ( { C }  u.  { D } ) )
32ineq2d 3199 . 2  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  { C ,  D } )  =  ( { A ,  B }  i^i  ( { C }  u.  { D } ) ) )
4 indi 3244 . . 3  |-  ( { A ,  B }  i^i  ( { C }  u.  { D } ) )  =  ( ( { A ,  B }  i^i  { C }
)  u.  ( { A ,  B }  i^i  { D } ) )
5 df-pr 3448 . . . . . . . 8  |-  { A ,  B }  =  ( { A }  u.  { B } )
65ineq1i 3195 . . . . . . 7  |-  ( { A ,  B }  i^i  { C } )  =  ( ( { A }  u.  { B } )  i^i  { C } )
7 indir 3246 . . . . . . 7  |-  ( ( { A }  u.  { B } )  i^i 
{ C } )  =  ( ( { A }  i^i  { C } )  u.  ( { B }  i^i  { C } ) )
86, 7eqtri 2108 . . . . . 6  |-  ( { A ,  B }  i^i  { C } )  =  ( ( { A }  i^i  { C } )  u.  ( { B }  i^i  { C } ) )
9 disjsn2 3500 . . . . . . . . . 10  |-  ( A  =/=  C  ->  ( { A }  i^i  { C } )  =  (/) )
109adantr 270 . . . . . . . . 9  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { A }  i^i  { C } )  =  (/) )
1110adantr 270 . . . . . . . 8  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A }  i^i  { C } )  =  (/) )
12 disjsn2 3500 . . . . . . . . . 10  |-  ( B  =/=  C  ->  ( { B }  i^i  { C } )  =  (/) )
1312adantl 271 . . . . . . . . 9  |-  ( ( A  =/=  C  /\  B  =/=  C )  -> 
( { B }  i^i  { C } )  =  (/) )
1413adantr 270 . . . . . . . 8  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { B }  i^i  { C } )  =  (/) )
1511, 14jca 300 . . . . . . 7  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A }  i^i  { C }
)  =  (/)  /\  ( { B }  i^i  { C } )  =  (/) ) )
16 un00 3326 . . . . . . 7  |-  ( ( ( { A }  i^i  { C } )  =  (/)  /\  ( { B }  i^i  { C } )  =  (/) ) 
<->  ( ( { A }  i^i  { C }
)  u.  ( { B }  i^i  { C } ) )  =  (/) )
1715, 16sylib 120 . . . . . 6  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A }  i^i  { C }
)  u.  ( { B }  i^i  { C } ) )  =  (/) )
188, 17syl5eq 2132 . . . . 5  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  { C } )  =  (/) )
195ineq1i 3195 . . . . . . 7  |-  ( { A ,  B }  i^i  { D } )  =  ( ( { A }  u.  { B } )  i^i  { D } )
20 indir 3246 . . . . . . 7  |-  ( ( { A }  u.  { B } )  i^i 
{ D } )  =  ( ( { A }  i^i  { D } )  u.  ( { B }  i^i  { D } ) )
2119, 20eqtri 2108 . . . . . 6  |-  ( { A ,  B }  i^i  { D } )  =  ( ( { A }  i^i  { D } )  u.  ( { B }  i^i  { D } ) )
22 disjsn2 3500 . . . . . . . . . 10  |-  ( A  =/=  D  ->  ( { A }  i^i  { D } )  =  (/) )
2322adantr 270 . . . . . . . . 9  |-  ( ( A  =/=  D  /\  B  =/=  D )  -> 
( { A }  i^i  { D } )  =  (/) )
2423adantl 271 . . . . . . . 8  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A }  i^i  { D } )  =  (/) )
25 disjsn2 3500 . . . . . . . . . 10  |-  ( B  =/=  D  ->  ( { B }  i^i  { D } )  =  (/) )
2625adantl 271 . . . . . . . . 9  |-  ( ( A  =/=  D  /\  B  =/=  D )  -> 
( { B }  i^i  { D } )  =  (/) )
2726adantl 271 . . . . . . . 8  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { B }  i^i  { D } )  =  (/) )
2824, 27jca 300 . . . . . . 7  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A }  i^i  { D }
)  =  (/)  /\  ( { B }  i^i  { D } )  =  (/) ) )
29 un00 3326 . . . . . . 7  |-  ( ( ( { A }  i^i  { D } )  =  (/)  /\  ( { B }  i^i  { D } )  =  (/) ) 
<->  ( ( { A }  i^i  { D }
)  u.  ( { B }  i^i  { D } ) )  =  (/) )
3028, 29sylib 120 . . . . . 6  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A }  i^i  { D }
)  u.  ( { B }  i^i  { D } ) )  =  (/) )
3121, 30syl5eq 2132 . . . . 5  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  { D } )  =  (/) )
3218, 31uneq12d 3153 . . . 4  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A ,  B }  i^i  { C } )  u.  ( { A ,  B }  i^i  { D } ) )  =  ( (/)  u.  (/) ) )
33 un0 3314 . . . 4  |-  ( (/)  u.  (/) )  =  (/)
3432, 33syl6eq 2136 . . 3  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A ,  B }  i^i  { C } )  u.  ( { A ,  B }  i^i  { D } ) )  =  (/) )
354, 34syl5eq 2132 . 2  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  ( { C }  u.  { D } ) )  =  (/) )
363, 35eqtrd 2120 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 102    = wceq 1289    =/= wne 2255    u. cun 2995    i^i cin 2996   (/)c0 3284   {csn 3441   {cpr 3442
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-sn 3447  df-pr 3448
This theorem is referenced by: (None)
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