Theorem snnex 4431

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 4119	. . . 4 ⊢ ¬ V ∈ V
2		vsnid 3613	. . . . . . . . 9 ⊢ 𝑧 ∈ {𝑧}
3		a9ev 1690	. . . . . . . . . 10 ⊢ ∃𝑦 𝑦 = 𝑧
4		sneq 3592	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → {𝑧} = {𝑦})
5	4	equcoms 1701	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → {𝑧} = {𝑦})
6	3, 5	eximii 1595	. . . . . . . . 9 ⊢ ∃𝑦{𝑧} = {𝑦}
7		vex 2733	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
8	7	snex 4169	. . . . . . . . . 10 ⊢ {𝑧} ∈ V
9		eleq2 2234	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑧} → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑧}))
10		eqeq1 2177	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑧} → (𝑥 = {𝑦} ↔ {𝑧} = {𝑦}))
11	10	exbidv 1818	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑧} → (∃𝑦 𝑥 = {𝑦} ↔ ∃𝑦{𝑧} = {𝑦}))
12	9, 11	anbi12d 470	. . . . . . . . . 10 ⊢ (𝑥 = {𝑧} → ((𝑧 ∈ 𝑥 ∧ ∃𝑦 𝑥 = {𝑦}) ↔ (𝑧 ∈ {𝑧} ∧ ∃𝑦{𝑧} = {𝑦})))
13	8, 12	spcev 2825	. . . . . . . . 9 ⊢ ((𝑧 ∈ {𝑧} ∧ ∃𝑦{𝑧} = {𝑦}) → ∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 𝑥 = {𝑦}))
14	2, 6, 13	mp2an 424	. . . . . . . 8 ⊢ ∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 𝑥 = {𝑦})
15		eluniab 3806	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ↔ ∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 𝑥 = {𝑦}))
16	14, 15	mpbir 145	. . . . . . 7 ⊢ 𝑧 ∈ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}}
17	16, 7	2th 173	. . . . . 6 ⊢ (𝑧 ∈ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ↔ 𝑧 ∈ V)
18	17	eqriv 2167	. . . . 5 ⊢ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} = V
19	18	eleq1i 2236	. . . 4 ⊢ (∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V ↔ V ∈ V)
20	1, 19	mtbir 666	. . 3 ⊢ ¬ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
21		uniexg 4422	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
22	20, 21	mto 657	. 2 ⊢ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
23	22	nelir 2438	1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V

Syntax hints:   ∧ wa 103   = wceq 1348  ∃wex 1485   ∈ wcel 2141  {cab 2156   ∉ wnel 2435  Vcvv 2730  {csn 3581  ∪ cuni 3794

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-un 4416

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-nel 2436  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-uni 3795

This theorem is referenced by:  fiprc  6789