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.

Step	Hyp	Ref	Expression
1		0ne1 10684	. . . . . . . 8 |- 0 =/= 1
2	1	neii 2628	. . . . . . 7 |- -. 0 = 1
3	2	intnanr 927	. . . . . 6 |- -. ( 0 = 1 /\ a = { 0 } )
4	3	intnanr 927	. . . . 5 |- -. ( ( 0 = 1 /\ a = { 0 } ) /\ ( ( 0 = 1 /\ b = { 0 , 1 } ) \/ ( 0 = 1 /\ b = { 0 , 1 } ) ) )
5	4	gen2 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
11	7, 8, 8, 8, 9, 10	propeqop 39009	. . . . . . . 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 265	. . . . . . 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 296	. . . . . 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 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 } ) ) ) ) )
15	14	albidv 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 } ) ) ) ) )
16	5, 15	mpbiri 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 >. )
18	16, 17	sylibr 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 >. )
20	18, 19	sylnibr 307	1 |- ( G = { <. 0 , 1 >. , <. 1 , 1 >. } -> -. G e. ( _V X. _V ) )

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)