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Theorem fun2dmnopgexmpl 39174
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 10699 . . . . . . . 8  |-  0  =/=  1
21neii 2645 . . . . . . 7  |-  -.  0  =  1
32intnanr 929 . . . . . 6  |-  -.  (
0  =  1  /\  a  =  { 0 } )
43intnanr 929 . . . . 5  |-  -.  (
( 0  =  1  /\  a  =  {
0 } )  /\  ( ( 0  =  1  /\  b  =  { 0 ,  1 } )  \/  (
0  =  1  /\  b  =  { 0 ,  1 } ) ) )
54gen2 1678 . . . 4  |-  A. a A. b  -.  (
( 0  =  1  /\  a  =  {
0 } )  /\  ( ( 0  =  1  /\  b  =  { 0 ,  1 } )  \/  (
0  =  1  /\  b  =  { 0 ,  1 } ) ) )
6 eqeq1 2475 . . . . . . . 8  |-  ( G  =  { <. 0 ,  1 >. ,  <. 1 ,  1 >. }  ->  ( G  = 
<. a ,  b >.  <->  {
<. 0 ,  1
>. ,  <. 1 ,  1 >. }  =  <. a ,  b >. )
)
7 c0ex 9655 . . . . . . . . 9  |-  0  e.  _V
8 1ex 9656 . . . . . . . . 9  |-  1  e.  _V
9 vex 3034 . . . . . . . . 9  |-  a  e. 
_V
10 vex 3034 . . . . . . . . 9  |-  b  e. 
_V
117, 8, 8, 8, 9, 10propeqop 39146 . . . . . . . 8  |-  ( {
<. 0 ,  1
>. ,  <. 1 ,  1 >. }  =  <. a ,  b >.  <->  ( (
0  =  1  /\  a  =  { 0 } )  /\  (
( 0  =  1  /\  b  =  {
0 ,  1 } )  \/  ( 0  =  1  /\  b  =  { 0 ,  1 } ) ) ) )
126, 11syl6bb 269 . . . . . . 7  |-  ( G  =  { <. 0 ,  1 >. ,  <. 1 ,  1 >. }  ->  ( G  = 
<. a ,  b >.  <->  ( ( 0  =  1  /\  a  =  {
0 } )  /\  ( ( 0  =  1  /\  b  =  { 0 ,  1 } )  \/  (
0  =  1  /\  b  =  { 0 ,  1 } ) ) ) ) )
1312notbid 301 . . . . . 6  |-  ( G  =  { <. 0 ,  1 >. ,  <. 1 ,  1 >. }  ->  ( -.  G  =  <. a ,  b
>. 
<->  -.  ( ( 0  =  1  /\  a  =  { 0 } )  /\  ( ( 0  =  1  /\  b  =  { 0 ,  1 } )  \/  (
0  =  1  /\  b  =  { 0 ,  1 } ) ) ) ) )
1413albidv 1775 . . . . 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 1775 . . . 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 241 . . 3  |-  ( G  =  { <. 0 ,  1 >. ,  <. 1 ,  1 >. }  ->  A. a A. b  -.  G  =  <. a ,  b >. )
17 2nexaln 1710 . . 3  |-  ( -. 
E. a E. b  G  =  <. a ,  b >.  <->  A. a A. b  -.  G  =  <. a ,  b >. )
1816, 17sylibr 217 . 2  |-  ( G  =  { <. 0 ,  1 >. ,  <. 1 ,  1 >. }  ->  -.  E. a E. b  G  =  <. a ,  b >.
)
19 elvv 4898 . 2  |-  ( G  e.  ( _V  X.  _V )  <->  E. a E. b  G  =  <. a ,  b >. )
2018, 19sylnibr 312 1  |-  ( G  =  { <. 0 ,  1 >. ,  <. 1 ,  1 >. }  ->  -.  G  e.  ( _V  X.  _V )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 375    /\ wa 376   A.wal 1450    = wceq 1452   E.wex 1671    e. wcel 1904   _Vcvv 3031   {csn 3959   {cpr 3961   <.cop 3965    X. cxp 4837   0cc0 9557   1c1 9558
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-mulcl 9619  ax-i2m1 9625  ax-1ne0 9626
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-opab 4455  df-xp 4845
This theorem is referenced by: (None)
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