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Theorem f1oun2prg 11864
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 10293 . . . . . . 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 10311 . . . . . . 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 9059 . . . . . . . 8  |-  1  =/=  0
1312necomi 2686 . . . . . . 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 5718 . . . 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 10133 . . . . . . . 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 10134 . . . . . . . 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 10069 . . . . . . . 8  |-  2  e.  RR
32 2lt3 10143 . . . . . . . 8  |-  2  <  3
3331, 32ltneii 9186 . . . . . . 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 5718 . . . 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 3869 . . . . . . . . . 10  |-  ( A  =/=  C  ->  ( { A }  i^i  { C } )  =  (/) )
40393ad2ant2 979 . . . . . . . . 9  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D )  ->  ( { A }  i^i  { C } )  =  (/) )
41 disjsn2 3869 . . . . . . . . . 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 3821 . . . . . . . . . 10  |-  { A ,  B }  =  ( { A }  u.  { B } )
4645ineq1i 3538 . . . . . . . . 9  |-  ( { A ,  B }  i^i  { C } )  =  ( ( { A }  u.  { B } )  i^i  { C } )
4746eqeq1i 2443 . . . . . . . 8  |-  ( ( { A ,  B }  i^i  { C }
)  =  (/)  <->  ( ( { A }  u.  { B } )  i^i  { C } )  =  (/) )
48 undisj1 3679 . . . . . . . 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 3869 . . . . . . . . . 10  |-  ( A  =/=  D  ->  ( { A }  i^i  { D } )  =  (/) )
52513ad2ant3 980 . . . . . . . . 9  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D )  ->  ( { A }  i^i  { D } )  =  (/) )
53 disjsn2 3869 . . . . . . . . . 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 3538 . . . . . . . . 9  |-  ( { A ,  B }  i^i  { D } )  =  ( ( { A }  u.  { B } )  i^i  { D } )
5857eqeq1i 2443 . . . . . . . 8  |-  ( ( { A ,  B }  i^i  { D }
)  =  (/)  <->  ( ( { A }  u.  { B } )  i^i  { D } )  =  (/) )
59 undisj1 3679 . . . . . . . 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 3680 . . . . . 6  |-  ( ( ( { A ,  B }  i^i  { C } )  =  (/)  /\  ( { A ,  B }  i^i  { D } )  =  (/) ) 
<->  ( { A ,  B }  i^i  ( { C }  u.  { D } ) )  =  (/) )
64 df-pr 3821 . . . . . . . . 9  |-  { C ,  D }  =  ( { C }  u.  { D } )
6564eqcomi 2440 . . . . . . . 8  |-  ( { C }  u.  { D } )  =  { C ,  D }
6665ineq2i 3539 . . . . . . 7  |-  ( { A ,  B }  i^i  ( { C }  u.  { D } ) )  =  ( { A ,  B }  i^i  { C ,  D } )
6766eqeq1i 2443 . . . . . 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 3821 . . . . . . . . 9  |-  { 0 ,  1 }  =  ( { 0 }  u.  { 1 } )
7170eqcomi 2440 . . . . . . . 8  |-  ( { 0 }  u.  {
1 } )  =  { 0 ,  1 }
7271ineq1i 3538 . . . . . . 7  |-  ( ( { 0 }  u.  { 1 } )  i^i 
{ 2 } )  =  ( { 0 ,  1 }  i^i  { 2 } )
73 2ne0 10083 . . . . . . . . . . 11  |-  2  =/=  0
7473necomi 2686 . . . . . . . . . 10  |-  0  =/=  2
75 disjsn2 3869 . . . . . . . . . 10  |-  ( 0  =/=  2  ->  ( { 0 }  i^i  { 2 } )  =  (/) )
7674, 75ax-mp 8 . . . . . . . . 9  |-  ( { 0 }  i^i  {
2 } )  =  (/)
77 1ne2 10187 . . . . . . . . . 10  |-  1  =/=  2
78 disjsn2 3869 . . . . . . . . . 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 3679 . . . . . . . 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 2458 . . . . . 6  |-  ( { 0 ,  1 }  i^i  { 2 } )  =  (/)
8471ineq1i 3538 . . . . . . 7  |-  ( ( { 0 }  u.  { 1 } )  i^i 
{ 3 } )  =  ( { 0 ,  1 }  i^i  { 3 } )
85 3ne0 10085 . . . . . . . . . . 11  |-  3  =/=  0
8685necomi 2686 . . . . . . . . . 10  |-  0  =/=  3
87 disjsn2 3869 . . . . . . . . . 10  |-  ( 0  =/=  3  ->  ( { 0 }  i^i  { 3 } )  =  (/) )
8886, 87ax-mp 8 . . . . . . . . 9  |-  ( { 0 }  i^i  {
3 } )  =  (/)
89 1re 9090 . . . . . . . . . . 11  |-  1  e.  RR
90 1lt3 10144 . . . . . . . . . . 11  |-  1  <  3
9189, 90ltneii 9186 . . . . . . . . . 10  |-  1  =/=  3
92 disjsn2 3869 . . . . . . . . . 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 3679 . . . . . . . 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 2458 . . . . . 6  |-  ( { 0 ,  1 }  i^i  { 3 } )  =  (/)
9883, 97pm3.2i 442 . . . . 5  |-  ( ( { 0 ,  1 }  i^i  { 2 } )  =  (/)  /\  ( { 0 ,  1 }  i^i  {
3 } )  =  (/) )
99 undisj2 3680 . . . . . 6  |-  ( ( ( { 0 ,  1 }  i^i  {
2 } )  =  (/)  /\  ( { 0 ,  1 }  i^i  { 3 } )  =  (/) )  <->  ( { 0 ,  1 }  i^i  ( { 2 }  u.  { 3 } ) )  =  (/) )
100 df-pr 3821 . . . . . . . . 9  |-  { 2 ,  3 }  =  ( { 2 }  u.  { 3 } )
101100eqcomi 2440 . . . . . . . 8  |-  ( { 2 }  u.  {
3 } )  =  { 2 ,  3 }
102101ineq2i 3539 . . . . . . 7  |-  ( { 0 ,  1 }  i^i  ( { 2 }  u.  { 3 } ) )  =  ( { 0 ,  1 }  i^i  {
2 ,  3 } )
103102eqeq1i 2443 . . . . . 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 5694 . . 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 1652    e. wcel 1725    =/= wne 2599    u. cun 3318    i^i cin 3319   (/)c0 3628   {csn 3814   {cpr 3815   <.cop 3817   -1-1-onto->wf1o 5453   0cc0 8990   1c1 8991   NNcn 10000   2c2 10049   3c3 10050   ZZcz 10282
This theorem is referenced by:  s4f1o  11865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-2 10058  df-3 10059  df-z 10283
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