Theorem snnex 4466

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 4150	. . . 4 ⊢ ¬ V ∈ V
2		vsnid 3639	. . . . . . . . 9 ⊢ 𝑧 ∈ {𝑧}
3		a9ev 1708	. . . . . . . . . 10 ⊢ ∃𝑦 𝑦 = 𝑧
4		sneq 3618	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → {𝑧} = {𝑦})
5	4	equcoms 1719	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → {𝑧} = {𝑦})
6	3, 5	eximii 1613	. . . . . . . . 9 ⊢ ∃𝑦{𝑧} = {𝑦}
7		vex 2755	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
8	7	snex 4203	. . . . . . . . . 10 ⊢ {𝑧} ∈ V
9		eleq2 2253	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑧} → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑧}))
10		eqeq1 2196	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑧} → (𝑥 = {𝑦} ↔ {𝑧} = {𝑦}))
11	10	exbidv 1836	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑧} → (∃𝑦 𝑥 = {𝑦} ↔ ∃𝑦{𝑧} = {𝑦}))
12	9, 11	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑥 = {𝑧} → ((𝑧 ∈ 𝑥 ∧ ∃𝑦 𝑥 = {𝑦}) ↔ (𝑧 ∈ {𝑧} ∧ ∃𝑦{𝑧} = {𝑦})))
13	8, 12	spcev 2847	. . . . . . . . 9 ⊢ ((𝑧 ∈ {𝑧} ∧ ∃𝑦{𝑧} = {𝑦}) → ∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 𝑥 = {𝑦}))
14	2, 6, 13	mp2an 426	. . . . . . . 8 ⊢ ∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 𝑥 = {𝑦})
15		eluniab 3836	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ↔ ∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 𝑥 = {𝑦}))
16	14, 15	mpbir 146	. . . . . . 7 ⊢ 𝑧 ∈ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}}
17	16, 7	2th 174	. . . . . 6 ⊢ (𝑧 ∈ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ↔ 𝑧 ∈ V)
18	17	eqriv 2186	. . . . 5 ⊢ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} = V
19	18	eleq1i 2255	. . . 4 ⊢ (∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V ↔ V ∈ V)
20	1, 19	mtbir 672	. . 3 ⊢ ¬ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
21		uniexg 4457	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
22	20, 21	mto 663	. 2 ⊢ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
23	22	nelir 2458	1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V

Syntax hints:   ∧ wa 104   = wceq 1364  ∃wex 1503   ∈ wcel 2160  {cab 2175   ∉ wnel 2455  Vcvv 2752  {csn 3607  ∪ cuni 3824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-un 4451

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-nel 2456  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-uni 3825

This theorem is referenced by:  fiprc  6841