Theorem snnexOLD 6967

Description: Obsolete proof of snnex 6966 as of 5-Dec-2021. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
snnexOLD	|- { x | E. y x = { y } } e/ _V

Distinct variable group:   x, y

Proof of Theorem snnexOLD

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vprc 4796	. . . 4 |- -. _V e. _V
2		vsnid 4209	. . . . . . . . 9 |- z e. { z }
3		ax6ev 1890	. . . . . . . . . 10 |- E. y y = z
4		sneq 4187	. . . . . . . . . . 11 |- ( z = y -> { z } = { y } )
5	4	equcoms 1947	. . . . . . . . . 10 |- ( y = z -> { z } = { y } )
6	3, 5	eximii 1764	. . . . . . . . 9 |- E. y { z } = { y }
7		snex 4908	. . . . . . . . . 10 |- { z } e. _V
8		eleq2 2690	. . . . . . . . . . 11 |- ( x = { z } -> ( z e. x <-> z e. { z } ) )
9		eqeq1 2626	. . . . . . . . . . . 12 |- ( x = { z } -> ( x = { y } <-> { z } = { y } ) )
10	9	exbidv 1850	. . . . . . . . . . 11 |- ( x = { z } -> ( E. y x = { y } <-> E. y { z } = { y } ) )
11	8, 10	anbi12d 747	. . . . . . . . . 10 |- ( x = { z } -> ( ( z e. x /\ E. y x = { y } ) <-> ( z e. { z } /\ E. y { z } = { y } ) ) )
12	7, 11	spcev 3300	. . . . . . . . 9 |- ( ( z e. { z } /\ E. y { z } = { y } ) -> E. x ( z e. x /\ E. y x = { y } ) )
13	2, 6, 12	mp2an 708	. . . . . . . 8 |- E. x ( z e. x /\ E. y x = { y } )
14		eluniab 4447	. . . . . . . 8 |- ( z e. U. { x | E. y x = { y } } <-> E. x ( z e. x /\ E. y x = { y } ) )
15	13, 14	mpbir 221	. . . . . . 7 |- z e. U. { x | E. y x = { y } }
16		vex 3203	. . . . . . 7 |- z e. _V
17	15, 16	2th 254	. . . . . 6 |- ( z e. U. { x | E. y x = { y } } <-> z e. _V )
18	17	eqriv 2619	. . . . 5 |- U. { x | E. y x = { y } } = _V
19	18	eleq1i 2692	. . . 4 |- ( U. { x | E. y x = { y } } e. _V <-> _V e. _V )
20	1, 19	mtbir 313	. . 3 |- -. U. { x | E. y x = { y } } e. _V
21		uniexg 6955	. . 3 |- ( { x | E. y x = { y } } e. _V -> U. { x | E. y x = { y } } e. _V )
22	20, 21	mto 188	. 2 |- -. { x | E. y x = { y } } e. _V
23	22	nelir 2900	1 |- { x | E. y x = { y } } e/ _V

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   {cab 2608   e/ wnel 2897   _Vcvv 3200   {csn 4177   U.cuni 4436

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-8 1992  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-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-nel 2898  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-uni 4437

This theorem is referenced by: (None)