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Theorem fun2dmnopgexmpl 39038
Description: A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 21-Sep-2020.)
Assertion
Ref Expression
fun2dmnopgexmpl  |-  ( G  =  { <. 0 ,  1 >. ,  <. 1 ,  1 >. }  ->  -.  G  e.  ( _V  X.  _V )
)

Proof of Theorem fun2dmnopgexmpl
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ne1 10684 . . . . . . . 8  |-  0  =/=  1
21neii 2628 . . . . . . 7  |-  -.  0  =  1
32intnanr 927 . . . . . 6  |-  -.  (
0  =  1  /\  a  =  { 0 } )
43intnanr 927 . . . . 5  |-  -.  (
( 0  =  1  /\  a  =  {
0 } )  /\  ( ( 0  =  1  /\  b  =  { 0 ,  1 } )  \/  (
0  =  1  /\  b  =  { 0 ,  1 } ) ) )
54gen2 1672 . . . 4  |-  A. a A. b  -.  (
( 0  =  1  /\  a  =  {
0 } )  /\  ( ( 0  =  1  /\  b  =  { 0 ,  1 } )  \/  (
0  =  1  /\  b  =  { 0 ,  1 } ) ) )
6 eqeq1 2457 . . . . . . . 8  |-  ( G  =  { <. 0 ,  1 >. ,  <. 1 ,  1 >. }  ->  ( G  = 
<. a ,  b >.  <->  {
<. 0 ,  1
>. ,  <. 1 ,  1 >. }  =  <. a ,  b >. )
)
7 c0ex 9642 . . . . . . . . 9  |-  0  e.  _V
8 1ex 9643 . . . . . . . . 9  |-  1  e.  _V
9 vex 3050 . . . . . . . . 9  |-  a  e. 
_V
10 vex 3050 . . . . . . . . 9  |-  b  e. 
_V
117, 8, 8, 8, 9, 10propeqop 39009 . . . . . . . 8  |-  ( {
<. 0 ,  1
>. ,  <. 1 ,  1 >. }  =  <. a ,  b >.  <->  ( (
0  =  1  /\  a  =  { 0 } )  /\  (
( 0  =  1  /\  b  =  {
0 ,  1 } )  \/  ( 0  =  1  /\  b  =  { 0 ,  1 } ) ) ) )
126, 11syl6bb 265 . . . . . . 7  |-  ( G  =  { <. 0 ,  1 >. ,  <. 1 ,  1 >. }  ->  ( G  = 
<. a ,  b >.  <->  ( ( 0  =  1  /\  a  =  {
0 } )  /\  ( ( 0  =  1  /\  b  =  { 0 ,  1 } )  \/  (
0  =  1  /\  b  =  { 0 ,  1 } ) ) ) ) )
1312notbid 296 . . . . . 6  |-  ( G  =  { <. 0 ,  1 >. ,  <. 1 ,  1 >. }  ->  ( -.  G  =  <. a ,  b
>. 
<->  -.  ( ( 0  =  1  /\  a  =  { 0 } )  /\  ( ( 0  =  1  /\  b  =  { 0 ,  1 } )  \/  (
0  =  1  /\  b  =  { 0 ,  1 } ) ) ) ) )
1413albidv 1769 . . . . 5  |-  ( G  =  { <. 0 ,  1 >. ,  <. 1 ,  1 >. }  ->  ( A. b  -.  G  =  <. a ,  b >.  <->  A. b  -.  ( ( 0  =  1  /\  a  =  { 0 } )  /\  ( ( 0  =  1  /\  b  =  { 0 ,  1 } )  \/  (
0  =  1  /\  b  =  { 0 ,  1 } ) ) ) ) )
1514albidv 1769 . . . 4  |-  ( G  =  { <. 0 ,  1 >. ,  <. 1 ,  1 >. }  ->  ( A. a A. b  -.  G  =  <. a ,  b
>. 
<-> 
A. a A. b  -.  ( ( 0  =  1  /\  a  =  { 0 } )  /\  ( ( 0  =  1  /\  b  =  { 0 ,  1 } )  \/  (
0  =  1  /\  b  =  { 0 ,  1 } ) ) ) ) )
165, 15mpbiri 237 . . 3  |-  ( G  =  { <. 0 ,  1 >. ,  <. 1 ,  1 >. }  ->  A. a A. b  -.  G  =  <. a ,  b >. )
17 2nexaln 1704 . . 3  |-  ( -. 
E. a E. b  G  =  <. a ,  b >.  <->  A. a A. b  -.  G  =  <. a ,  b >. )
1816, 17sylibr 216 . 2  |-  ( G  =  { <. 0 ,  1 >. ,  <. 1 ,  1 >. }  ->  -.  E. a E. b  G  =  <. a ,  b >.
)
19 elvv 4896 . 2  |-  ( G  e.  ( _V  X.  _V )  <->  E. a E. b  G  =  <. a ,  b >. )
2018, 19sylnibr 307 1  |-  ( G  =  { <. 0 ,  1 >. ,  <. 1 ,  1 >. }  ->  -.  G  e.  ( _V  X.  _V )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 370    /\ wa 371   A.wal 1444    = wceq 1446   E.wex 1665    e. wcel 1889   _Vcvv 3047   {csn 3970   {cpr 3972   <.cop 3976    X. cxp 4835   0cc0 9544   1c1 9545
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pr 4642  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-mulcl 9606  ax-i2m1 9612  ax-1ne0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-v 3049  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-sn 3971  df-pr 3973  df-op 3977  df-opab 4465  df-xp 4843
This theorem is referenced by: (None)
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