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Theorem disjpr2 3458
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 3407 . . . 4  |-  { C ,  D }  =  ( { C }  u.  { D } )
21a1i 9 . . 3  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  ->  { C ,  D }  =  ( { C }  u.  { D } ) )
32ineq2d 3168 . 2  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  { C ,  D } )  =  ( { A ,  B }  i^i  ( { C }  u.  { D } ) ) )
4 indi 3212 . . 3  |-  ( { A ,  B }  i^i  ( { C }  u.  { D } ) )  =  ( ( { A ,  B }  i^i  { C }
)  u.  ( { A ,  B }  i^i  { D } ) )
5 df-pr 3407 . . . . . . . 8  |-  { A ,  B }  =  ( { A }  u.  { B } )
65ineq1i 3164 . . . . . . 7  |-  ( { A ,  B }  i^i  { C } )  =  ( ( { A }  u.  { B } )  i^i  { C } )
7 indir 3214 . . . . . . 7  |-  ( ( { A }  u.  { B } )  i^i 
{ C } )  =  ( ( { A }  i^i  { C } )  u.  ( { B }  i^i  { C } ) )
86, 7eqtri 2102 . . . . . 6  |-  ( { A ,  B }  i^i  { C } )  =  ( ( { A }  i^i  { C } )  u.  ( { B }  i^i  { C } ) )
9 disjsn2 3457 . . . . . . . . . 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 3457 . . . . . . . . . 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 3291 . . . . . . 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 2126 . . . . 5  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  { C } )  =  (/) )
195ineq1i 3164 . . . . . . 7  |-  ( { A ,  B }  i^i  { D } )  =  ( ( { A }  u.  { B } )  i^i  { D } )
20 indir 3214 . . . . . . 7  |-  ( ( { A }  u.  { B } )  i^i 
{ D } )  =  ( ( { A }  i^i  { D } )  u.  ( { B }  i^i  { D } ) )
2119, 20eqtri 2102 . . . . . 6  |-  ( { A ,  B }  i^i  { D } )  =  ( ( { A }  i^i  { D } )  u.  ( { B }  i^i  { D } ) )
22 disjsn2 3457 . . . . . . . . . 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 3457 . . . . . . . . . 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 3291 . . . . . . 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 2126 . . . . 5  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  { D } )  =  (/) )
3218, 31uneq12d 3128 . . . 4  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A ,  B }  i^i  { C } )  u.  ( { A ,  B }  i^i  { D } ) )  =  ( (/)  u.  (/) ) )
33 un0 3279 . . . 4  |-  ( (/)  u.  (/) )  =  (/)
3432, 33syl6eq 2130 . . 3  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( ( { A ,  B }  i^i  { C } )  u.  ( { A ,  B }  i^i  { D } ) )  =  (/) )
354, 34syl5eq 2126 . 2  |-  ( ( ( A  =/=  C  /\  B  =/=  C
)  /\  ( A  =/=  D  /\  B  =/= 
D ) )  -> 
( { A ,  B }  i^i  ( { C }  u.  { D } ) )  =  (/) )
363, 35eqtrd 2114 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 1285    =/= wne 2246    u. cun 2972    i^i cin 2973   (/)c0 3252   {csn 3400   {cpr 3401
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-v 2604  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-sn 3406  df-pr 3407
This theorem is referenced by: (None)
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