MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  f1oun2prg Structured version   Unicode version

Theorem f1oun2prg 11856
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 10285 . . . . . . 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 10303 . . . . . . 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 9051 . . . . . . . 8  |-  1  =/=  0
1312necomi 2680 . . . . . . 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 5710 . . . 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 10125 . . . . . . . 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 10126 . . . . . . . 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 10061 . . . . . . . 8  |-  2  e.  RR
32 2lt3 10135 . . . . . . . 8  |-  2  <  3
3331, 32ltneii 9178 . . . . . . 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 5710 . . . 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 3861 . . . . . . . . . 10  |-  ( A  =/=  C  ->  ( { A }  i^i  { C } )  =  (/) )
40393ad2ant2 979 . . . . . . . . 9  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D )  ->  ( { A }  i^i  { C } )  =  (/) )
41 disjsn2 3861 . . . . . . . . . 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 3813 . . . . . . . . . 10  |-  { A ,  B }  =  ( { A }  u.  { B } )
4645ineq1i 3530 . . . . . . . . 9  |-  ( { A ,  B }  i^i  { C } )  =  ( ( { A }  u.  { B } )  i^i  { C } )
4746eqeq1i 2442 . . . . . . . 8  |-  ( ( { A ,  B }  i^i  { C }
)  =  (/)  <->  ( ( { A }  u.  { B } )  i^i  { C } )  =  (/) )
48 undisj1 3671 . . . . . . . 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 3861 . . . . . . . . . 10  |-  ( A  =/=  D  ->  ( { A }  i^i  { D } )  =  (/) )
52513ad2ant3 980 . . . . . . . . 9  |-  ( ( A  =/=  B  /\  A  =/=  C  /\  A  =/=  D )  ->  ( { A }  i^i  { D } )  =  (/) )
53 disjsn2 3861 . . . . . . . . . 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 3530 . . . . . . . . 9  |-  ( { A ,  B }  i^i  { D } )  =  ( ( { A }  u.  { B } )  i^i  { D } )
5857eqeq1i 2442 . . . . . . . 8  |-  ( ( { A ,  B }  i^i  { D }
)  =  (/)  <->  ( ( { A }  u.  { B } )  i^i  { D } )  =  (/) )
59 undisj1 3671 . . . . . . . 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 3672 . . . . . 6  |-  ( ( ( { A ,  B }  i^i  { C } )  =  (/)  /\  ( { A ,  B }  i^i  { D } )  =  (/) ) 
<->  ( { A ,  B }  i^i  ( { C }  u.  { D } ) )  =  (/) )
64 df-pr 3813 . . . . . . . . 9  |-  { C ,  D }  =  ( { C }  u.  { D } )
6564eqcomi 2439 . . . . . . . 8  |-  ( { C }  u.  { D } )  =  { C ,  D }
6665ineq2i 3531 . . . . . . 7  |-  ( { A ,  B }  i^i  ( { C }  u.  { D } ) )  =  ( { A ,  B }  i^i  { C ,  D } )
6766eqeq1i 2442 . . . . . 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 3813 . . . . . . . . 9  |-  { 0 ,  1 }  =  ( { 0 }  u.  { 1 } )
7170eqcomi 2439 . . . . . . . 8  |-  ( { 0 }  u.  {
1 } )  =  { 0 ,  1 }
7271ineq1i 3530 . . . . . . 7  |-  ( ( { 0 }  u.  { 1 } )  i^i 
{ 2 } )  =  ( { 0 ,  1 }  i^i  { 2 } )
73 2ne0 10075 . . . . . . . . . . 11  |-  2  =/=  0
7473necomi 2680 . . . . . . . . . 10  |-  0  =/=  2
75 disjsn2 3861 . . . . . . . . . 10  |-  ( 0  =/=  2  ->  ( { 0 }  i^i  { 2 } )  =  (/) )
7674, 75ax-mp 8 . . . . . . . . 9  |-  ( { 0 }  i^i  {
2 } )  =  (/)
77 1ne2 10179 . . . . . . . . . 10  |-  1  =/=  2
78 disjsn2 3861 . . . . . . . . . 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 3671 . . . . . . . 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 2457 . . . . . 6  |-  ( { 0 ,  1 }  i^i  { 2 } )  =  (/)
8471ineq1i 3530 . . . . . . 7  |-  ( ( { 0 }  u.  { 1 } )  i^i 
{ 3 } )  =  ( { 0 ,  1 }  i^i  { 3 } )
85 3ne0 10077 . . . . . . . . . . 11  |-  3  =/=  0
8685necomi 2680 . . . . . . . . . 10  |-  0  =/=  3
87 disjsn2 3861 . . . . . . . . . 10  |-  ( 0  =/=  3  ->  ( { 0 }  i^i  { 3 } )  =  (/) )
8886, 87ax-mp 8 . . . . . . . . 9  |-  ( { 0 }  i^i  {
3 } )  =  (/)
89 1re 9082 . . . . . . . . . . 11  |-  1  e.  RR
90 1lt3 10136 . . . . . . . . . . 11  |-  1  <  3
9189, 90ltneii 9178 . . . . . . . . . 10  |-  1  =/=  3
92 disjsn2 3861 . . . . . . . . . 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 3671 . . . . . . . 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 2457 . . . . . 6  |-  ( { 0 ,  1 }  i^i  { 3 } )  =  (/)
9883, 97pm3.2i 442 . . . . 5  |-  ( ( { 0 ,  1 }  i^i  { 2 } )  =  (/)  /\  ( { 0 ,  1 }  i^i  {
3 } )  =  (/) )
99 undisj2 3672 . . . . . 6  |-  ( ( ( { 0 ,  1 }  i^i  {
2 } )  =  (/)  /\  ( { 0 ,  1 }  i^i  { 3 } )  =  (/) )  <->  ( { 0 ,  1 }  i^i  ( { 2 }  u.  { 3 } ) )  =  (/) )
100 df-pr 3813 . . . . . . . . 9  |-  { 2 ,  3 }  =  ( { 2 }  u.  { 3 } )
101100eqcomi 2439 . . . . . . . 8  |-  ( { 2 }  u.  {
3 } )  =  { 2 ,  3 }
102101ineq2i 3531 . . . . . . 7  |-  ( { 0 ,  1 }  i^i  ( { 2 }  u.  { 3 } ) )  =  ( { 0 ,  1 }  i^i  {
2 ,  3 } )
103102eqeq1i 2442 . . . . . 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 5686 . . 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 2598    u. cun 3310    i^i cin 3311   (/)c0 3620   {csn 3806   {cpr 3807   <.cop 3809   -1-1-onto->wf1o 5445   0cc0 8982   1c1 8983   NNcn 9992   2c2 10041   3c3 10042   ZZcz 10274
This theorem is referenced by:  s4f1o  11857
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 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-recs 6625  df-rdg 6660  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-z 10275
  Copyright terms: Public domain W3C validator