Theorem snnex 4127

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 3858	. . . 4 ⊢ ¬ V ∈ V
2		vex 2534	. . . . . . . . . 10 ⊢ z ∈ V
3	2	snid 3373	. . . . . . . . 9 ⊢ z ∈ {z}
4		a9ev 1565	. . . . . . . . . 10 ⊢ ∃y y = z
5		sneq 3357	. . . . . . . . . . 11 ⊢ (z = y → {z} = {y})
6	5	equcoms 1572	. . . . . . . . . 10 ⊢ (y = z → {z} = {y})
7	4, 6	eximii 1471	. . . . . . . . 9 ⊢ ∃y{z} = {y}
8		snexgOLD 3905	. . . . . . . . . . 11 ⊢ (z ∈ V → {z} ∈ V)
9	2, 8	ax-mp 7	. . . . . . . . . 10 ⊢ {z} ∈ V
10		eleq2 2079	. . . . . . . . . . 11 ⊢ (x = {z} → (z ∈ x ↔ z ∈ {z}))
11		eqeq1 2024	. . . . . . . . . . . 12 ⊢ (x = {z} → (x = {y} ↔ {z} = {y}))
12	11	exbidv 1684	. . . . . . . . . . 11 ⊢ (x = {z} → (∃y x = {y} ↔ ∃y{z} = {y}))
13	10, 12	anbi12d 445	. . . . . . . . . 10 ⊢ (x = {z} → ((z ∈ x ∧ ∃y x = {y}) ↔ (z ∈ {z} ∧ ∃y{z} = {y})))
14	9, 13	spcev 2620	. . . . . . . . 9 ⊢ ((z ∈ {z} ∧ ∃y{z} = {y}) → ∃x(z ∈ x ∧ ∃y x = {y}))
15	3, 7, 14	mp2an 404	. . . . . . . 8 ⊢ ∃x(z ∈ x ∧ ∃y x = {y})
16		eluniab 3562	. . . . . . . 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 2015	. . . . 5 ⊢ ∪ {x ∣ ∃y x = {y}} = V
20	19	eleq1i 2081	. . . 4 ⊢ (∪ {x ∣ ∃y x = {y}} ∈ V ↔ V ∈ V)
21	1, 20	mtbir 583	. . 3 ⊢ ¬ ∪ {x ∣ ∃y x = {y}} ∈ V
22		uniexg 4121	. . 3 ⊢ ({x ∣ ∃y x = {y}} ∈ V → ∪ {x ∣ ∃y x = {y}} ∈ V)
23	21, 22	mto 575	. 2 ⊢ ¬ {x ∣ ∃y x = {y}} ∈ V
24	23	nelir 2274	1 ⊢ {x ∣ ∃y x = {y}} ∉ V

Syntax hints:   ∧ wa 97   = wceq 1226  ∃wex 1358   ∈ wcel 1370  {cab 2004   ∉ wnel 2183  Vcvv 2531  {csn 3346  ∪ cuni 3550

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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-un 4116

This theorem depends on definitions:  df-bi 110  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-nel 2185  df-rex 2286  df-v 2533  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-uni 3551

This theorem is referenced by: (None)