Theorem fun2dmnopgexmpl 41303

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 11088	. . . . . . . 8 |- 0 =/= 1
2	1	neii 2796	. . . . . . 7 |- -. 0 = 1
3	2	intnanr 961	. . . . . 6 |- -. ( 0 = 1 /\ a = { 0 } )
4	3	intnanr 961	. . . . 5 |- -. ( ( 0 = 1 /\ a = { 0 } ) /\ ( ( 0 = 1 /\ b = { 0 , 1 } ) \/ ( 0 = 1 /\ b = { 0 , 1 } ) ) )
5	4	gen2 1723	. . . 4 |- A. a A. b -. ( ( 0 = 1 /\ a = { 0 } ) /\ ( ( 0 = 1 /\ b = { 0 , 1 } ) \/ ( 0 = 1 /\ b = { 0 , 1 } ) ) )
6		eqeq1 2626	. . . . . . . 8 |- ( G = { <. 0 , 1 >. , <. 1 , 1 >. } -> ( G = <. a , b >. <-> { <. 0 , 1 >. , <. 1 , 1 >. } = <. a , b >. ) )
7		c0ex 10034	. . . . . . . . 9 |- 0 e. _V
8		1ex 10035	. . . . . . . . 9 |- 1 e. _V
9		vex 3203	. . . . . . . . 9 |- a e. _V
10		vex 3203	. . . . . . . . 9 |- b e. _V
11	7, 8, 8, 8, 9, 10	propeqop 4970	. . . . . . . 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 276	. . . . . . 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 308	. . . . . 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 1849	. . . . 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 1849	. . . 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 248	. . 3 |- ( G = { <. 0 , 1 >. , <. 1 , 1 >. } -> A. a A. b -. G = <. a , b >. )
17		2nexaln 1757	. . 3 |- ( -. E. a E. b G = <. a , b >. <-> A. a A. b -. G = <. a , b >. )
18	16, 17	sylibr 224	. 2 |- ( G = { <. 0 , 1 >. , <. 1 , 1 >. } -> -. E. a E. b G = <. a , b >. )
19		elvv 5177	. 2 |- ( G e. ( _V X. _V ) <-> E. a E. b G = <. a , b >. )
20	18, 19	sylnibr 319	1 |- ( G = { <. 0 , 1 >. , <. 1 , 1 >. } -> -. G e. ( _V X. _V ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   {csn 4177   {cpr 4179   <.cop 4183   X. cxp 5112   0cc0 9936   1c1 9937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-i2m1 10004  ax-1ne0 10005

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120

This theorem is referenced by: (None)