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Theorem f1oun2prg 11819
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 444 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W )  ->  A  e.  V )
2 0z 10249 . . . . . . 7  |-  0  e.  ZZ
31, 2jctil 524 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 0  e.  ZZ  /\  A  e.  V ) )
43ad2antrr 707 . . . . 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 ) )
5 simpr 448 . . . . . . 7  |-  ( ( A  e.  V  /\  B  e.  W )  ->  B  e.  W )
6 1z 10267 . . . . . . 7  |-  1  e.  ZZ
75, 6jctil 524 . . . . . 6  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( 1  e.  ZZ  /\  B  e.  W ) )
87ad2antrr 707 . . . . 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 ) )
94, 8jca 519 . . . 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 )
) )
10 id 20 . . . . . . . 8  |-  ( A  =/=  B  ->  A  =/=  B )
11103ad2ant1 978 . . . . . . 7  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D )  ->  A  =/=  B )
12 ax-1ne0 9015 . . . . . . . 8  |-  1  =/=  0
1312necomi 2649 . . . . . . 7  |-  0  =/=  1
1411, 13jctil 524 . . . . . 6  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D )  ->  (
0  =/=  1  /\  A  =/=  B ) )
1514adantr 452 . . . . 5  |-  ( ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D
)  /\  ( B  =/=  C  /\  B  =/= 
D  /\  C  =/=  D ) )  ->  (
0  =/=  1  /\  A  =/=  B ) )
1615adantl 453 . . . 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 ) )
17 f1oprg 5677 . . . 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 } ) )
189, 16, 17sylc 58 . . 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 }
)
19 simpl 444 . . . . . . . 8  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  C  e.  X )
20 2nn 10089 . . . . . . . 8  |-  2  e.  NN
2119, 20jctil 524 . . . . . . 7  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  ( 2  e.  NN  /\  C  e.  X ) )
2221adantl 453 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( 2  e.  NN  /\  C  e.  X ) )
23 simpr 448 . . . . . . . 8  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  D  e.  Y )
24 3nn 10090 . . . . . . . 8  |-  3  e.  NN
2523, 24jctil 524 . . . . . . 7  |-  ( ( C  e.  X  /\  D  e.  Y )  ->  ( 3  e.  NN  /\  D  e.  Y ) )
2625adantl 453 . . . . . 6  |-  ( ( ( A  e.  V  /\  B  e.  W
)  /\  ( C  e.  X  /\  D  e.  Y ) )  -> 
( 3  e.  NN  /\  D  e.  Y ) )
2722, 26jca 519 . . . . 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 )
) )
2827adantr 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  e.  NN  /\  C  e.  X )  /\  (
3  e.  NN  /\  D  e.  Y )
) )
29 id 20 . . . . . . . 8  |-  ( C  =/=  D  ->  C  =/=  D )
30293ad2ant3 980 . . . . . . 7  |-  ( ( B  =/=  C  /\  B  =/=  D  /\  C  =/=  D )  ->  C  =/=  D )
31 2re 10025 . . . . . . . 8  |-  2  e.  RR
32 2lt3 10099 . . . . . . . 8  |-  2  <  3
3331, 32ltneii 9142 . . . . . . 7  |-  2  =/=  3
3430, 33jctil 524 . . . . . 6  |-  ( ( B  =/=  C  /\  B  =/=  D  /\  C  =/=  D )  ->  (
2  =/=  3  /\  C  =/=  D ) )
3534adantl 453 . . . . 5  |-  ( ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D
)  /\  ( B  =/=  C  /\  B  =/= 
D  /\  C  =/=  D ) )  ->  (
2  =/=  3  /\  C  =/=  D ) )
3635adantl 453 . . . 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 ) )
37 f1oprg 5677 . . . 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 } ) )
3828, 36, 37sylc 58 . . 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 }
)
39 disjsn2 3829 . . . . . . . . . 10  |-  ( A  =/=  C  ->  ( { A }  i^i  { C } )  =  (/) )
40393ad2ant2 979 . . . . . . . . 9  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D )  ->  ( { A }  i^i  { C } )  =  (/) )
41 disjsn2 3829 . . . . . . . . . 10  |-  ( B  =/=  C  ->  ( { B }  i^i  { C } )  =  (/) )
42413ad2ant1 978 . . . . . . . . 9  |-  ( ( B  =/=  C  /\  B  =/=  D  /\  C  =/=  D )  ->  ( { B }  i^i  { C } )  =  (/) )
4340, 42anim12i 550 . . . . . . . 8  |-  ( ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D
)  /\  ( B  =/=  C  /\  B  =/= 
D  /\  C  =/=  D ) )  ->  (
( { A }  i^i  { C } )  =  (/)  /\  ( { B }  i^i  { C } )  =  (/) ) )
4443adantl 453 . . . . . . 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 } )  =  (/) ) )
45 df-pr 3781 . . . . . . . . . 10  |-  { A ,  B }  =  ( { A }  u.  { B } )
4645ineq1i 3498 . . . . . . . . 9  |-  ( { A ,  B }  i^i  { C } )  =  ( ( { A }  u.  { B } )  i^i  { C } )
4746eqeq1i 2411 . . . . . . . 8  |-  ( ( { A ,  B }  i^i  { C }
)  =  (/)  <->  ( ( { A }  u.  { B } )  i^i  { C } )  =  (/) )
48 undisj1 3639 . . . . . . . 8  |-  ( ( ( { A }  i^i  { C } )  =  (/)  /\  ( { B }  i^i  { C } )  =  (/) ) 
<->  ( ( { A }  u.  { B } )  i^i  { C } )  =  (/) )
4947, 48bitr4i 244 . . . . . . 7  |-  ( ( { A ,  B }  i^i  { C }
)  =  (/)  <->  ( ( { A }  i^i  { C } )  =  (/)  /\  ( { B }  i^i  { C } )  =  (/) ) )
5044, 49sylibr 204 . . . . . 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 } )  =  (/) )
51 disjsn2 3829 . . . . . . . . . 10  |-  ( A  =/=  D  ->  ( { A }  i^i  { D } )  =  (/) )
52513ad2ant3 980 . . . . . . . . 9  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D )  ->  ( { A }  i^i  { D } )  =  (/) )
53 disjsn2 3829 . . . . . . . . . 10  |-  ( B  =/=  D  ->  ( { B }  i^i  { D } )  =  (/) )
54533ad2ant2 979 . . . . . . . . 9  |-  ( ( B  =/=  C  /\  B  =/=  D  /\  C  =/=  D )  ->  ( { B }  i^i  { D } )  =  (/) )
5552, 54anim12i 550 . . . . . . . 8  |-  ( ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D
)  /\  ( B  =/=  C  /\  B  =/= 
D  /\  C  =/=  D ) )  ->  (
( { A }  i^i  { D } )  =  (/)  /\  ( { B }  i^i  { D } )  =  (/) ) )
5655adantl 453 . . . . . . 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 } )  =  (/) ) )
5745ineq1i 3498 . . . . . . . . 9  |-  ( { A ,  B }  i^i  { D } )  =  ( ( { A }  u.  { B } )  i^i  { D } )
5857eqeq1i 2411 . . . . . . . 8  |-  ( ( { A ,  B }  i^i  { D }
)  =  (/)  <->  ( ( { A }  u.  { B } )  i^i  { D } )  =  (/) )
59 undisj1 3639 . . . . . . . 8  |-  ( ( ( { A }  i^i  { D } )  =  (/)  /\  ( { B }  i^i  { D } )  =  (/) ) 
<->  ( ( { A }  u.  { B } )  i^i  { D } )  =  (/) )
6058, 59bitr4i 244 . . . . . . 7  |-  ( ( { A ,  B }  i^i  { D }
)  =  (/)  <->  ( ( { A }  i^i  { D } )  =  (/)  /\  ( { B }  i^i  { D } )  =  (/) ) )
6156, 60sylibr 204 . . . . . 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 } )  =  (/) )
6250, 61jca 519 . . . . 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 } )  =  (/) ) )
63 undisj2 3640 . . . . . 6  |-  ( ( ( { A ,  B }  i^i  { C } )  =  (/)  /\  ( { A ,  B }  i^i  { D } )  =  (/) ) 
<->  ( { A ,  B }  i^i  ( { C }  u.  { D } ) )  =  (/) )
64 df-pr 3781 . . . . . . . . 9  |-  { C ,  D }  =  ( { C }  u.  { D } )
6564eqcomi 2408 . . . . . . . 8  |-  ( { C }  u.  { D } )  =  { C ,  D }
6665ineq2i 3499 . . . . . . 7  |-  ( { A ,  B }  i^i  ( { C }  u.  { D } ) )  =  ( { A ,  B }  i^i  { C ,  D } )
6766eqeq1i 2411 . . . . . 6  |-  ( ( { A ,  B }  i^i  ( { C }  u.  { D } ) )  =  (/) 
<->  ( { A ,  B }  i^i  { C ,  D } )  =  (/) )
6863, 67bitri 241 . . . . 5  |-  ( ( ( { A ,  B }  i^i  { C } )  =  (/)  /\  ( { A ,  B }  i^i  { D } )  =  (/) ) 
<->  ( { A ,  B }  i^i  { C ,  D } )  =  (/) )
6962, 68sylib 189 . . . 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 }
)  =  (/) )
70 df-pr 3781 . . . . . . . . 9  |-  { 0 ,  1 }  =  ( { 0 }  u.  { 1 } )
7170eqcomi 2408 . . . . . . . 8  |-  ( { 0 }  u.  {
1 } )  =  { 0 ,  1 }
7271ineq1i 3498 . . . . . . 7  |-  ( ( { 0 }  u.  { 1 } )  i^i 
{ 2 } )  =  ( { 0 ,  1 }  i^i  { 2 } )
73 2ne0 10039 . . . . . . . . . . 11  |-  2  =/=  0
7473necomi 2649 . . . . . . . . . 10  |-  0  =/=  2
75 disjsn2 3829 . . . . . . . . . 10  |-  ( 0  =/=  2  ->  ( { 0 }  i^i  { 2 } )  =  (/) )
7674, 75ax-mp 8 . . . . . . . . 9  |-  ( { 0 }  i^i  {
2 } )  =  (/)
77 1ne2 10143 . . . . . . . . . 10  |-  1  =/=  2
78 disjsn2 3829 . . . . . . . . . 10  |-  ( 1  =/=  2  ->  ( { 1 }  i^i  { 2 } )  =  (/) )
7977, 78ax-mp 8 . . . . . . . . 9  |-  ( { 1 }  i^i  {
2 } )  =  (/)
8076, 79pm3.2i 442 . . . . . . . 8  |-  ( ( { 0 }  i^i  { 2 } )  =  (/)  /\  ( { 1 }  i^i  { 2 } )  =  (/) )
81 undisj1 3639 . . . . . . . 8  |-  ( ( ( { 0 }  i^i  { 2 } )  =  (/)  /\  ( { 1 }  i^i  { 2 } )  =  (/) )  <->  ( ( { 0 }  u.  {
1 } )  i^i 
{ 2 } )  =  (/) )
8280, 81mpbi 200 . . . . . . 7  |-  ( ( { 0 }  u.  { 1 } )  i^i 
{ 2 } )  =  (/)
8372, 82eqtr3i 2426 . . . . . 6  |-  ( { 0 ,  1 }  i^i  { 2 } )  =  (/)
8471ineq1i 3498 . . . . . . 7  |-  ( ( { 0 }  u.  { 1 } )  i^i 
{ 3 } )  =  ( { 0 ,  1 }  i^i  { 3 } )
85 3ne0 10041 . . . . . . . . . . 11  |-  3  =/=  0
8685necomi 2649 . . . . . . . . . 10  |-  0  =/=  3
87 disjsn2 3829 . . . . . . . . . 10  |-  ( 0  =/=  3  ->  ( { 0 }  i^i  { 3 } )  =  (/) )
8886, 87ax-mp 8 . . . . . . . . 9  |-  ( { 0 }  i^i  {
3 } )  =  (/)
89 1re 9046 . . . . . . . . . . 11  |-  1  e.  RR
90 1lt3 10100 . . . . . . . . . . 11  |-  1  <  3
9189, 90ltneii 9142 . . . . . . . . . 10  |-  1  =/=  3
92 disjsn2 3829 . . . . . . . . . 10  |-  ( 1  =/=  3  ->  ( { 1 }  i^i  { 3 } )  =  (/) )
9391, 92ax-mp 8 . . . . . . . . 9  |-  ( { 1 }  i^i  {
3 } )  =  (/)
9488, 93pm3.2i 442 . . . . . . . 8  |-  ( ( { 0 }  i^i  { 3 } )  =  (/)  /\  ( { 1 }  i^i  { 3 } )  =  (/) )
95 undisj1 3639 . . . . . . . 8  |-  ( ( ( { 0 }  i^i  { 3 } )  =  (/)  /\  ( { 1 }  i^i  { 3 } )  =  (/) )  <->  ( ( { 0 }  u.  {
1 } )  i^i 
{ 3 } )  =  (/) )
9694, 95mpbi 200 . . . . . . 7  |-  ( ( { 0 }  u.  { 1 } )  i^i 
{ 3 } )  =  (/)
9784, 96eqtr3i 2426 . . . . . 6  |-  ( { 0 ,  1 }  i^i  { 3 } )  =  (/)
9883, 97pm3.2i 442 . . . . 5  |-  ( ( { 0 ,  1 }  i^i  { 2 } )  =  (/)  /\  ( { 0 ,  1 }  i^i  {
3 } )  =  (/) )
99 undisj2 3640 . . . . . 6  |-  ( ( ( { 0 ,  1 }  i^i  {
2 } )  =  (/)  /\  ( { 0 ,  1 }  i^i  { 3 } )  =  (/) )  <->  ( { 0 ,  1 }  i^i  ( { 2 }  u.  { 3 } ) )  =  (/) )
100 df-pr 3781 . . . . . . . . 9  |-  { 2 ,  3 }  =  ( { 2 }  u.  { 3 } )
101100eqcomi 2408 . . . . . . . 8  |-  ( { 2 }  u.  {
3 } )  =  { 2 ,  3 }
102101ineq2i 3499 . . . . . . 7  |-  ( { 0 ,  1 }  i^i  ( { 2 }  u.  { 3 } ) )  =  ( { 0 ,  1 }  i^i  {
2 ,  3 } )
103102eqeq1i 2411 . . . . . 6  |-  ( ( { 0 ,  1 }  i^i  ( { 2 }  u.  {
3 } ) )  =  (/)  <->  ( { 0 ,  1 }  i^i  { 2 ,  3 } )  =  (/) )
10499, 103bitri 241 . . . . 5  |-  ( ( ( { 0 ,  1 }  i^i  {
2 } )  =  (/)  /\  ( { 0 ,  1 }  i^i  { 3 } )  =  (/) )  <->  ( { 0 ,  1 }  i^i  { 2 ,  3 } )  =  (/) )
10598, 104mpbi 200 . . . 4  |-  ( { 0 ,  1 }  i^i  { 2 ,  3 } )  =  (/)
10669, 105jctil 524 . . 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 }
)  =  (/) ) )
107 f1oun 5653 . . 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 } ) )
10818, 38, 106, 107syl21anc 1183 . 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 } ) )
109108ex 424 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567    u. cun 3278    i^i cin 3279   (/)c0 3588   {csn 3774   {cpr 3775   <.cop 3777   -1-1-onto->wf1o 5412   0cc0 8946   1c1 8947   NNcn 9956   2c2 10005   3c3 10006   ZZcz 10238
This theorem is referenced by:  s4f1o  11820
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-recs 6592  df-rdg 6627  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-z 10239
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