Theorem snnex 4147

Description: The class of all singletons is a proper class. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.)

Assertion

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

Distinct variable group:   x,y

Proof of Theorem snnex

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vprc 3879	. . . 4 |- -. _V e. _V
2		vex 2554	. . . . . . . . . 10 |- z e. _V
3	2	snid 3394	. . . . . . . . 9 |- z e. {z }
4		a9ev 1584	. . . . . . . . . 10 |- E.y y = z
5		sneq 3378	. . . . . . . . . . 11 |- (z = y -> {z } = {y })
6	5	equcoms 1591	. . . . . . . . . 10 |- (y = z -> {z } = {y })
7	4, 6	eximii 1490	. . . . . . . . 9 |- E.y {z } = {y }
8		snexgOLD 3926	. . . . . . . . . . 11 |- (z e. _V -> {z } e. _V)
9	2, 8	ax-mp 7	. . . . . . . . . 10 |- {z } e. _V
10		eleq2 2098	. . . . . . . . . . 11 |- (x = {z } -> (z e. x <-> z e. {z }))
11		eqeq1 2043	. . . . . . . . . . . 12 |- (x = {z } -> (x = {y } <-> {z } = {y }))
12	11	exbidv 1703	. . . . . . . . . . 11 |- (x = {z } -> (E.y x = {y } <-> E.y {z } = {y }))
13	10, 12	anbi12d 442	. . . . . . . . . 10 |- (x = {z } -> ((z e. x /\ E.y x = {y }) <-> (z e. {z } /\ E.y {z } = {y })))
14	9, 13	spcev 2641	. . . . . . . . 9 |- ((z e. {z } /\ E.y {z } = {y }) -> E.x(z e. x /\ E.y x = {y }))
15	3, 7, 14	mp2an 402	. . . . . . . 8 |- E.x(z e. x /\ E.y x = {y })
16		eluniab 3583	. . . . . . . 8 |- (z e. U. {x | E.y x = {y } } <-> E.x(z e. x /\ E.y x = {y }))
17	15, 16	mpbir 134	. . . . . . 7 |- z e. U. {x | E.y x = {y } }
18	17, 2	2th 163	. . . . . 6 |- (z e. U. {x | E.y x = {y } } <-> z e. _V)
19	18	eqriv 2034	. . . . 5 |- U. {x | E.y x = {y } } = _V
20	19	eleq1i 2100	. . . 4 |- ( U. {x | E.y x = {y } } e. _V <-> _V e. _V)
21	1, 20	mtbir 595	. . 3 |- -. U. {x | E.y x = {y } } e. _V
22		uniexg 4141	. . 3 |- ( {x | E.y x = {y } } e. _V -> U. {x | E.y x = {y } } e. _V)
23	21, 22	mto 587	. 2 |- -. {x | E.y x = {y } } e. _V
24	23	nelir 2294	1 |- {x | E.y x = {y } } e/ _V

Syntax hints:   /\ wa 97   = wceq 1242  E.wex 1378   e. wcel 1390   {cab 2023   e/ wnel 2202   _Vcvv 2551   {csn 3367   U.cuni 3571

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-nel 2204  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-uni 3572

This theorem is referenced by:  fiprc  6228