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 ∣ ∃y x = {y}} ∉ 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 ∈ V
2		vex 2554	. . . . . . . . . 10 ⊢ z ∈ V
3	2	snid 3394	. . . . . . . . 9 ⊢ z ∈ {z}
4		a9ev 1584	. . . . . . . . . 10 ⊢ ∃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 ⊢ ∃y{z} = {y}
8		snexgOLD 3926	. . . . . . . . . . 11 ⊢ (z ∈ V → {z} ∈ V)
9	2, 8	ax-mp 7	. . . . . . . . . 10 ⊢ {z} ∈ V
10		eleq2 2098	. . . . . . . . . . 11 ⊢ (x = {z} → (z ∈ x ↔ z ∈ {z}))
11		eqeq1 2043	. . . . . . . . . . . 12 ⊢ (x = {z} → (x = {y} ↔ {z} = {y}))
12	11	exbidv 1703	. . . . . . . . . . 11 ⊢ (x = {z} → (∃y x = {y} ↔ ∃y{z} = {y}))
13	10, 12	anbi12d 442	. . . . . . . . . 10 ⊢ (x = {z} → ((z ∈ x ∧ ∃y x = {y}) ↔ (z ∈ {z} ∧ ∃y{z} = {y})))
14	9, 13	spcev 2641	. . . . . . . . 9 ⊢ ((z ∈ {z} ∧ ∃y{z} = {y}) → ∃x(z ∈ x ∧ ∃y x = {y}))
15	3, 7, 14	mp2an 402	. . . . . . . 8 ⊢ ∃x(z ∈ x ∧ ∃y x = {y})
16		eluniab 3583	. . . . . . . 8 ⊢ (z ∈ ∪ {x ∣ ∃y x = {y}} ↔ ∃x(z ∈ x ∧ ∃y x = {y}))
17	15, 16	mpbir 134	. . . . . . 7 ⊢ z ∈ ∪ {x ∣ ∃y x = {y}}
18	17, 2	2th 163	. . . . . 6 ⊢ (z ∈ ∪ {x ∣ ∃y x = {y}} ↔ z ∈ V)
19	18	eqriv 2034	. . . . 5 ⊢ ∪ {x ∣ ∃y x = {y}} = V
20	19	eleq1i 2100	. . . 4 ⊢ (∪ {x ∣ ∃y x = {y}} ∈ V ↔ V ∈ V)
21	1, 20	mtbir 595	. . 3 ⊢ ¬ ∪ {x ∣ ∃y x = {y}} ∈ V
22		uniexg 4141	. . 3 ⊢ ({x ∣ ∃y x = {y}} ∈ V → ∪ {x ∣ ∃y x = {y}} ∈ V)
23	21, 22	mto 587	. 2 ⊢ ¬ {x ∣ ∃y x = {y}} ∈ V
24	23	nelir 2294	1 ⊢ {x ∣ ∃y x = {y}} ∉ V

Syntax hints:   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  {cab 2023   ∉ wnel 2202  Vcvv 2551  {csn 3367  ∪ 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