Theorem snnex 6862

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 4724	. . . 4 ⊢ ¬ V ∈ V
2		vsnid 4156	. . . . . . . . 9 ⊢ 𝑧 ∈ {𝑧}
3		ax6ev 1877	. . . . . . . . . 10 ⊢ ∃𝑦 𝑦 = 𝑧
4		sneq 4135	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → {𝑧} = {𝑦})
5	4	equcoms 1934	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → {𝑧} = {𝑦})
6	3, 5	eximii 1754	. . . . . . . . 9 ⊢ ∃𝑦{𝑧} = {𝑦}
7		snex 4835	. . . . . . . . . 10 ⊢ {𝑧} ∈ V
8		eleq2 2677	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑧} → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑧}))
9		eqeq1 2614	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑧} → (𝑥 = {𝑦} ↔ {𝑧} = {𝑦}))
10	9	exbidv 1837	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑧} → (∃𝑦 𝑥 = {𝑦} ↔ ∃𝑦{𝑧} = {𝑦}))
11	8, 10	anbi12d 743	. . . . . . . . . 10 ⊢ (𝑥 = {𝑧} → ((𝑧 ∈ 𝑥 ∧ ∃𝑦 𝑥 = {𝑦}) ↔ (𝑧 ∈ {𝑧} ∧ ∃𝑦{𝑧} = {𝑦})))
12	7, 11	spcev 3273	. . . . . . . . 9 ⊢ ((𝑧 ∈ {𝑧} ∧ ∃𝑦{𝑧} = {𝑦}) → ∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 𝑥 = {𝑦}))
13	2, 6, 12	mp2an 704	. . . . . . . 8 ⊢ ∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 𝑥 = {𝑦})
14		eluniab 4383	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ↔ ∃𝑥(𝑧 ∈ 𝑥 ∧ ∃𝑦 𝑥 = {𝑦}))
15	13, 14	mpbir 220	. . . . . . 7 ⊢ 𝑧 ∈ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}}
16		vex 3176	. . . . . . 7 ⊢ 𝑧 ∈ V
17	15, 16	2th 253	. . . . . 6 ⊢ (𝑧 ∈ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ↔ 𝑧 ∈ V)
18	17	eqriv 2607	. . . . 5 ⊢ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} = V
19	18	eleq1i 2679	. . . 4 ⊢ (∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V ↔ V ∈ V)
20	1, 19	mtbir 312	. . 3 ⊢ ¬ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
21		uniexg 6853	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
22	20, 21	mto 187	. 2 ⊢ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
23	22	nelir 2886	1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V

Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ∉ wnel 2781  Vcvv 3173  {csn 4125  ∪ cuni 4372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-nel 2783  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373

This theorem is referenced by:  fiprc  7924