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.

Step	Hyp	Ref	Expression
1		0ne1 10699	. . . . . . . 8 |- 0 =/= 1
2	1	neii 2645	. . . . . . 7 |- -. 0 = 1
3	2	intnanr 929	. . . . . 6 |- -. ( 0 = 1 /\ a = { 0 } )
4	3	intnanr 929	. . . . 5 |- -. ( ( 0 = 1 /\ a = { 0 } ) /\ ( ( 0 = 1 /\ b = { 0 , 1 } ) \/ ( 0 = 1 /\ b = { 0 , 1 } ) ) )
5	4	gen2 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
11	7, 8, 8, 8, 9, 10	propeqop 39146	. . . . . . . 8 |- ( { <. 0 , 1 >. , <. 1 , 1 >. } = <. a , b >. <-> ( ( 0 = 1 /\ a = { 0 } ) /\ ( ( 0 = 1 /\ b = { 0 , 1 } ) \/ ( 0 = 1 /\ b = { 0 , 1 } ) ) ) )
12	6, 11	syl6bb 269	. . . . . . 7 |- ( G = { <. 0 , 1 >. , <. 1 , 1 >. } -> ( G = <. a , b >. <-> ( ( 0 = 1 /\ a = { 0 } ) /\ ( ( 0 = 1 /\ b = { 0 , 1 } ) \/ ( 0 = 1 /\ b = { 0 , 1 } ) ) ) ) )
13	12	notbid 301	. . . . . 6 |- ( G = { <. 0 , 1 >. , <. 1 , 1 >. } -> ( -. G = <. a , b >. <-> -. ( ( 0 = 1 /\ a = { 0 } ) /\ ( ( 0 = 1 /\ b = { 0 , 1 } ) \/ ( 0 = 1 /\ b = { 0 , 1 } ) ) ) ) )
14	13	albidv 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 } ) ) ) ) )
15	14	albidv 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 } ) ) ) ) )
16	5, 15	mpbiri 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 >. )
18	16, 17	sylibr 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 >. )
20	18, 19	sylnibr 312	1 |- ( G = { <. 0 , 1 >. , <. 1 , 1 >. } -> -. G e. ( _V X. _V ) )

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)