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Theorem f1oun2prg 28087
Description: A union of unordered pairs of ordered pairs with different elements is a one-to-one onto function. (Contributed by Alexander van der Vekens, 14-Aug-2017.)
Assertion
Ref Expression
f1oun2prg  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) )  -> 
( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
<. 3 ,  D >. } ) : ( { 0 ,  1 }  u.  { 2 ,  3 } ) -1-1-onto-> ( { A ,  B }  u.  { C ,  D } ) ) )

Proof of Theorem f1oun2prg
StepHypRef Expression
1 simpl 443 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  e.  V )
2 0z 10037 . . . . . . . 8  |-  0  e.  ZZ
31, 2jctil 523 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 0  e.  ZZ  /\  A  e.  V ) )
43adantr 451 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( 0  e.  ZZ  /\  A  e.  V ) )
54adantr 451 . . . . 5  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) )  ->  ( 0  e.  ZZ  /\  A  e.  V ) )
6 simpr 447 . . . . . . . 8  |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  e.  W )
7 1z 10055 . . . . . . . 8  |-  1  e.  ZZ
86, 7jctil 523 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 1  e.  ZZ  /\  B  e.  W ) )
98adantr 451 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( 1  e.  ZZ  /\  B  e.  W ) )
109adantr 451 . . . . 5  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) )  ->  ( 1  e.  ZZ  /\  B  e.  W ) )
115, 10jca 518 . . . 4  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) )  ->  ( ( 0  e.  ZZ  /\  A  e.  V )  /\  (
1  e.  ZZ  /\  B  e.  W )
) )
12 id 19 . . . . . . . 8  |-  ( A  =/=  B  ->  A  =/=  B )
13123ad2ant1 976 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D )  ->  A  =/=  B )
14 ax-1ne0 8808 . . . . . . . 8  |-  1  =/=  0
1514necomi 2530 . . . . . . 7  |-  0  =/=  1
1613, 15jctil 523 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D )  ->  (
0  =/=  1  /\  A  =/=  B ) )
1716adantr 451 . . . . 5  |-  ( ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D
)  /\  ( B  =/=  C  /\  B  =/= 
D  /\  C  =/=  D ) )  ->  (
0  =/=  1  /\  A  =/=  B ) )
1817adantl 452 . . . 4  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) )  ->  ( 0  =/=  1  /\  A  =/= 
B ) )
19 f1oprg 28086 . . . 4  |-  ( ( ( 0  e.  ZZ  /\  A  e.  V )  /\  ( 1  e.  ZZ  /\  B  e.  W ) )  -> 
( ( 0  =/=  1  /\  A  =/= 
B )  ->  { <. 0 ,  A >. , 
<. 1 ,  B >. } : { 0 ,  1 } -1-1-onto-> { A ,  B } ) )
2011, 18, 19sylc 56 . . 3  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) )  ->  { <. 0 ,  A >. ,  <. 1 ,  B >. } : {
0 ,  1 } -1-1-onto-> { A ,  B }
)
21 simpl 443 . . . . . . . 8  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  C  e.  X )
22 2nn 9879 . . . . . . . 8  |-  2  e.  NN
2321, 22jctil 523 . . . . . . 7  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  ( 2  e.  NN  /\  C  e.  X ) )
2423adantl 452 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( 2  e.  NN  /\  C  e.  X ) )
25 simpr 447 . . . . . . . 8  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  D  e.  Y )
26 3nn 9880 . . . . . . . 8  |-  3  e.  NN
2725, 26jctil 523 . . . . . . 7  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  ( 3  e.  NN  /\  D  e.  Y ) )
2827adantl 452 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( 3  e.  NN  /\  D  e.  Y ) )
2924, 28jca 518 . . . . 5  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( ( 2  e.  NN  /\  C  e.  X )  /\  (
3  e.  NN  /\  D  e.  Y )
) )
3029adantr 451 . . . 4  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) )  ->  ( ( 2  e.  NN  /\  C  e.  X )  /\  (
3  e.  NN  /\  D  e.  Y )
) )
31 id 19 . . . . . . . 8  |-  ( C  =/=  D  ->  C  =/=  D )
32313ad2ant3 978 . . . . . . 7  |-  ( ( B  =/=  C  /\  B  =/=  D  /\  C  =/=  D )  ->  C  =/=  D )
33 2re 9817 . . . . . . . 8  |-  2  e.  RR
34 2lt3 9889 . . . . . . . 8  |-  2  <  3
3533, 34ltneii 8933 . . . . . . 7  |-  2  =/=  3
3632, 35jctil 523 . . . . . 6  |-  ( ( B  =/=  C  /\  B  =/=  D  /\  C  =/=  D )  ->  (
2  =/=  3  /\  C  =/=  D ) )
3736adantl 452 . . . . 5  |-  ( ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D
)  /\  ( B  =/=  C  /\  B  =/= 
D  /\  C  =/=  D ) )  ->  (
2  =/=  3  /\  C  =/=  D ) )
3837adantl 452 . . . 4  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) )  ->  ( 2  =/=  3  /\  C  =/= 
D ) )
39 f1oprg 28086 . . . 4  |-  ( ( ( 2  e.  NN  /\  C  e.  X )  /\  ( 3  e.  NN  /\  D  e.  Y ) )  -> 
( ( 2  =/=  3  /\  C  =/= 
D )  ->  { <. 2 ,  C >. , 
<. 3 ,  D >. } : { 2 ,  3 } -1-1-onto-> { C ,  D } ) )
4030, 38, 39sylc 56 . . 3  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) )  ->  { <. 2 ,  C >. ,  <. 3 ,  D >. } : {
2 ,  3 } -1-1-onto-> { C ,  D }
)
41 disjsn2 3696 . . . . . . . . . 10  |-  ( A  =/=  C  ->  ( { A }  i^i  { C } )  =  (/) )
42413ad2ant2 977 . . . . . . . . 9  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D )  ->  ( { A }  i^i  { C } )  =  (/) )
43 disjsn2 3696 . . . . . . . . . 10  |-  ( B  =/=  C  ->  ( { B }  i^i  { C } )  =  (/) )
44433ad2ant1 976 . . . . . . . . 9  |-  ( ( B  =/=  C  /\  B  =/=  D  /\  C  =/=  D )  ->  ( { B }  i^i  { C } )  =  (/) )
4542, 44anim12i 549 . . . . . . . 8  |-  ( ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D
)  /\  ( B  =/=  C  /\  B  =/= 
D  /\  C  =/=  D ) )  ->  (
( { A }  i^i  { C } )  =  (/)  /\  ( { B }  i^i  { C } )  =  (/) ) )
4645adantl 452 . . . . . . 7  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) )  ->  ( ( { A }  i^i  { C } )  =  (/)  /\  ( { B }  i^i  { C } )  =  (/) ) )
47 df-pr 3649 . . . . . . . . . 10  |-  { A ,  B }  =  ( { A }  u.  { B } )
4847ineq1i 3368 . . . . . . . . 9  |-  ( { A ,  B }  i^i  { C } )  =  ( ( { A }  u.  { B } )  i^i  { C } )
4948eqeq1i 2292 . . . . . . . 8  |-  ( ( { A ,  B }  i^i  { C }
)  =  (/)  <->  ( ( { A }  u.  { B } )  i^i  { C } )  =  (/) )
50 undisj1 3508 . . . . . . . 8  |-  ( ( ( { A }  i^i  { C } )  =  (/)  /\  ( { B }  i^i  { C } )  =  (/) ) 
<->  ( ( { A }  u.  { B } )  i^i  { C } )  =  (/) )
5149, 50bitr4i 243 . . . . . . 7  |-  ( ( { A ,  B }  i^i  { C }
)  =  (/)  <->  ( ( { A }  i^i  { C } )  =  (/)  /\  ( { B }  i^i  { C } )  =  (/) ) )
5246, 51sylibr 203 . . . . . 6  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) )  ->  ( { A ,  B }  i^i  { C } )  =  (/) )
53 disjsn2 3696 . . . . . . . . . 10  |-  ( A  =/=  D  ->  ( { A }  i^i  { D } )  =  (/) )
54533ad2ant3 978 . . . . . . . . 9  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D )  ->  ( { A }  i^i  { D } )  =  (/) )
55 disjsn2 3696 . . . . . . . . . 10  |-  ( B  =/=  D  ->  ( { B }  i^i  { D } )  =  (/) )
56553ad2ant2 977 . . . . . . . . 9  |-  ( ( B  =/=  C  /\  B  =/=  D  /\  C  =/=  D )  ->  ( { B }  i^i  { D } )  =  (/) )
5754, 56anim12i 549 . . . . . . . 8  |-  ( ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D
)  /\  ( B  =/=  C  /\  B  =/= 
D  /\  C  =/=  D ) )  ->  (
( { A }  i^i  { D } )  =  (/)  /\  ( { B }  i^i  { D } )  =  (/) ) )
5857adantl 452 . . . . . . 7  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) )  ->  ( ( { A }  i^i  { D } )  =  (/)  /\  ( { B }  i^i  { D } )  =  (/) ) )
5947ineq1i 3368 . . . . . . . . 9  |-  ( { A ,  B }  i^i  { D } )  =  ( ( { A }  u.  { B } )  i^i  { D } )
6059eqeq1i 2292 . . . . . . . 8  |-  ( ( { A ,  B }  i^i  { D }
)  =  (/)  <->  ( ( { A }  u.  { B } )  i^i  { D } )  =  (/) )
61 undisj1 3508 . . . . . . . 8  |-  ( ( ( { A }  i^i  { D } )  =  (/)  /\  ( { B }  i^i  { D } )  =  (/) ) 
<->  ( ( { A }  u.  { B } )  i^i  { D } )  =  (/) )
6260, 61bitr4i 243 . . . . . . 7  |-  ( ( { A ,  B }  i^i  { D }
)  =  (/)  <->  ( ( { A }  i^i  { D } )  =  (/)  /\  ( { B }  i^i  { D } )  =  (/) ) )
6358, 62sylibr 203 . . . . . 6  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) )  ->  ( { A ,  B }  i^i  { D } )  =  (/) )
6452, 63jca 518 . . . . 5  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) )  ->  ( ( { A ,  B }  i^i  { C } )  =  (/)  /\  ( { A ,  B }  i^i  { D } )  =  (/) ) )
65 undisj2 3509 . . . . . . 7  |-  ( ( ( { A ,  B }  i^i  { C } )  =  (/)  /\  ( { A ,  B }  i^i  { D } )  =  (/) ) 
<->  ( { A ,  B }  i^i  ( { C }  u.  { D } ) )  =  (/) )
66 df-pr 3649 . . . . . . . . . 10  |-  { C ,  D }  =  ( { C }  u.  { D } )
6766eqcomi 2289 . . . . . . . . 9  |-  ( { C }  u.  { D } )  =  { C ,  D }
6867ineq2i 3369 . . . . . . . 8  |-  ( { A ,  B }  i^i  ( { C }  u.  { D } ) )  =  ( { A ,  B }  i^i  { C ,  D } )
6968eqeq1i 2292 . . . . . . 7  |-  ( ( { A ,  B }  i^i  ( { C }  u.  { D } ) )  =  (/) 
<->  ( { A ,  B }  i^i  { C ,  D } )  =  (/) )
7065, 69bitri 240 . . . . . 6  |-  ( ( ( { A ,  B }  i^i  { C } )  =  (/)  /\  ( { A ,  B }  i^i  { D } )  =  (/) ) 
<->  ( { A ,  B }  i^i  { C ,  D } )  =  (/) )
7170bicomi 193 . . . . 5  |-  ( ( { A ,  B }  i^i  { C ,  D } )  =  (/)  <->  (
( { A ,  B }  i^i  { C } )  =  (/)  /\  ( { A ,  B }  i^i  { D } )  =  (/) ) )
7264, 71sylibr 203 . . . 4  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) )  ->  ( { A ,  B }  i^i  { C ,  D }
)  =  (/) )
73 df-pr 3649 . . . . . . . . 9  |-  { 0 ,  1 }  =  ( { 0 }  u.  { 1 } )
7473eqcomi 2289 . . . . . . . 8  |-  ( { 0 }  u.  {
1 } )  =  { 0 ,  1 }
7574ineq1i 3368 . . . . . . 7  |-  ( ( { 0 }  u.  { 1 } )  i^i 
{ 2 } )  =  ( { 0 ,  1 }  i^i  { 2 } )
76 2ne0 9831 . . . . . . . . . . 11  |-  2  =/=  0
7776necomi 2530 . . . . . . . . . 10  |-  0  =/=  2
78 disjsn2 3696 . . . . . . . . . 10  |-  ( 0  =/=  2  ->  ( { 0 }  i^i  { 2 } )  =  (/) )
7977, 78ax-mp 8 . . . . . . . . 9  |-  ( { 0 }  i^i  {
2 } )  =  (/)
80 1ne2 9933 . . . . . . . . . 10  |-  1  =/=  2
81 disjsn2 3696 . . . . . . . . . 10  |-  ( 1  =/=  2  ->  ( { 1 }  i^i  { 2 } )  =  (/) )
8280, 81ax-mp 8 . . . . . . . . 9  |-  ( { 1 }  i^i  {
2 } )  =  (/)
8379, 82pm3.2i 441 . . . . . . . 8  |-  ( ( { 0 }  i^i  { 2 } )  =  (/)  /\  ( { 1 }  i^i  { 2 } )  =  (/) )
84 undisj1 3508 . . . . . . . 8  |-  ( ( ( { 0 }  i^i  { 2 } )  =  (/)  /\  ( { 1 }  i^i  { 2 } )  =  (/) )  <->  ( ( { 0 }  u.  {
1 } )  i^i 
{ 2 } )  =  (/) )
8583, 84mpbi 199 . . . . . . 7  |-  ( ( { 0 }  u.  { 1 } )  i^i 
{ 2 } )  =  (/)
8675, 85eqtr3i 2307 . . . . . 6  |-  ( { 0 ,  1 }  i^i  { 2 } )  =  (/)
8774ineq1i 3368 . . . . . . 7  |-  ( ( { 0 }  u.  { 1 } )  i^i 
{ 3 } )  =  ( { 0 ,  1 }  i^i  { 3 } )
88 3ne0 9833 . . . . . . . . . . 11  |-  3  =/=  0
8988necomi 2530 . . . . . . . . . 10  |-  0  =/=  3
90 disjsn2 3696 . . . . . . . . . 10  |-  ( 0  =/=  3  ->  ( { 0 }  i^i  { 3 } )  =  (/) )
9189, 90ax-mp 8 . . . . . . . . 9  |-  ( { 0 }  i^i  {
3 } )  =  (/)
92 1re 8839 . . . . . . . . . . 11  |-  1  e.  RR
93 1lt3 9890 . . . . . . . . . . 11  |-  1  <  3
9492, 93ltneii 8933 . . . . . . . . . 10  |-  1  =/=  3
95 disjsn2 3696 . . . . . . . . . 10  |-  ( 1  =/=  3  ->  ( { 1 }  i^i  { 3 } )  =  (/) )
9694, 95ax-mp 8 . . . . . . . . 9  |-  ( { 1 }  i^i  {
3 } )  =  (/)
9791, 96pm3.2i 441 . . . . . . . 8  |-  ( ( { 0 }  i^i  { 3 } )  =  (/)  /\  ( { 1 }  i^i  { 3 } )  =  (/) )
98 undisj1 3508 . . . . . . . 8  |-  ( ( ( { 0 }  i^i  { 3 } )  =  (/)  /\  ( { 1 }  i^i  { 3 } )  =  (/) )  <->  ( ( { 0 }  u.  {
1 } )  i^i 
{ 3 } )  =  (/) )
9997, 98mpbi 199 . . . . . . 7  |-  ( ( { 0 }  u.  { 1 } )  i^i 
{ 3 } )  =  (/)
10087, 99eqtr3i 2307 . . . . . 6  |-  ( { 0 ,  1 }  i^i  { 3 } )  =  (/)
10186, 100pm3.2i 441 . . . . 5  |-  ( ( { 0 ,  1 }  i^i  { 2 } )  =  (/)  /\  ( { 0 ,  1 }  i^i  {
3 } )  =  (/) )
102 undisj2 3509 . . . . . 6  |-  ( ( ( { 0 ,  1 }  i^i  {
2 } )  =  (/)  /\  ( { 0 ,  1 }  i^i  { 3 } )  =  (/) )  <->  ( { 0 ,  1 }  i^i  ( { 2 }  u.  { 3 } ) )  =  (/) )
103 df-pr 3649 . . . . . . . . 9  |-  { 2 ,  3 }  =  ( { 2 }  u.  { 3 } )
104103eqcomi 2289 . . . . . . . 8  |-  ( { 2 }  u.  {
3 } )  =  { 2 ,  3 }
105104ineq2i 3369 . . . . . . 7  |-  ( { 0 ,  1 }  i^i  ( { 2 }  u.  { 3 } ) )  =  ( { 0 ,  1 }  i^i  {
2 ,  3 } )
106105eqeq1i 2292 . . . . . 6  |-  ( ( { 0 ,  1 }  i^i  ( { 2 }  u.  {
3 } ) )  =  (/)  <->  ( { 0 ,  1 }  i^i  { 2 ,  3 } )  =  (/) )
107102, 106bitri 240 . . . . 5  |-  ( ( ( { 0 ,  1 }  i^i  {
2 } )  =  (/)  /\  ( { 0 ,  1 }  i^i  { 3 } )  =  (/) )  <->  ( { 0 ,  1 }  i^i  { 2 ,  3 } )  =  (/) )
108101, 107mpbi 199 . . . 4  |-  ( { 0 ,  1 }  i^i  { 2 ,  3 } )  =  (/)
10972, 108jctil 523 . . 3  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) )  ->  ( ( { 0 ,  1 }  i^i  { 2 ,  3 } )  =  (/)  /\  ( { A ,  B }  i^i  { C ,  D }
)  =  (/) ) )
110 f1oun 5494 . . 3  |-  ( ( ( { <. 0 ,  A >. ,  <. 1 ,  B >. } : {
0 ,  1 } -1-1-onto-> { A ,  B }  /\  { <. 2 ,  C >. ,  <. 3 ,  D >. } : { 2 ,  3 } -1-1-onto-> { C ,  D } )  /\  (
( { 0 ,  1 }  i^i  {
2 ,  3 } )  =  (/)  /\  ( { A ,  B }  i^i  { C ,  D } )  =  (/) ) )  ->  ( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
<. 3 ,  D >. } ) : ( { 0 ,  1 }  u.  { 2 ,  3 } ) -1-1-onto-> ( { A ,  B }  u.  { C ,  D } ) )
11120, 40, 109, 110syl21anc 1181 . 2  |-  ( ( ( ( A  e.  V  /\  B  e.  W )  /\  ( C  e.  X  /\  D  e.  Y )
)  /\  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) ) )  ->  ( { <. 0 ,  A >. , 
<. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
<. 3 ,  D >. } ) : ( { 0 ,  1 }  u.  { 2 ,  3 } ) -1-1-onto-> ( { A ,  B }  u.  { C ,  D } ) )
112111ex 423 1  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/= 
D )  /\  ( B  =/=  C  /\  B  =/=  D  /\  C  =/= 
D ) )  -> 
( { <. 0 ,  A >. ,  <. 1 ,  B >. }  u.  { <. 2 ,  C >. , 
<. 3 ,  D >. } ) : ( { 0 ,  1 }  u.  { 2 ,  3 } ) -1-1-onto-> ( { A ,  B }  u.  { C ,  D } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448    u. cun 3152    i^i cin 3153   (/)c0 3457   {csn 3642   {cpr 3643   <.cop 3645   -1-1-onto->wf1o 5256   0cc0 8739   1c1 8740   NNcn 9748   2c2 9797   3c3 9798   ZZcz 10026
This theorem is referenced by:  s4f1o  28104
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816
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 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-riota 6306  df-recs 6390  df-rdg 6425  df-er 6662  df-en 6866  df-dom 6867  df-sdom 6868  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-z 10027
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