Theorem snnex 4181

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 3888	. . . 4 ⊢ ¬ V ∈ V
2		vex 2560	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
3	2	snid 3402	. . . . . . . . 9 ⊢ 𝑧 ∈ {𝑧}
4		a9ev 1587	. . . . . . . . . 10 ⊢ ∃𝑦 𝑦 = 𝑧
5		sneq 3386	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → {𝑧} = {𝑦})
6	5	equcoms 1594	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → {𝑧} = {𝑦})
7	4, 6	eximii 1493	. . . . . . . . 9 ⊢ ∃𝑦{𝑧} = {𝑦}
8		snexgOLD 3935	. . . . . . . . . . 11 ⊢ (𝑧 ∈ V → {𝑧} ∈ V)
9	2, 8	ax-mp 7	. . . . . . . . . 10 ⊢ {𝑧} ∈ V
10		eleq2 2101	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑧} → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑧}))
11		eqeq1 2046	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑧} → (𝑥 = {𝑦} ↔ {𝑧} = {𝑦}))
12	11	exbidv 1706	. . . . . . . . . . 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 2647	. . . . . . . . 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 3592	. . . . . . . 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 2037	. . . . 5 ⊢ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} = V
20	19	eleq1i 2103	. . . 4 ⊢ (∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V ↔ V ∈ V)
21	1, 20	mtbir 596	. . 3 ⊢ ¬ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
22		uniexg 4175	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
23	21, 22	mto 588	. 2 ⊢ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
24	23	nelir 2300	1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V

Syntax hints:   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393  {cab 2026   ∉ wnel 2205  Vcvv 2557  {csn 3375  ∪ cuni 3580

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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-nel 2207  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-uni 3581

This theorem is referenced by:  fiprc  6292