Theorem snnex 4503

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 4184	. . . 4 ⊢ ¬ V ∈ V
2		vsnid 3670	. . . . . . . . 9 ⊢ 𝑧 ∈ {𝑧}
3		a9ev 1721	. . . . . . . . . 10 ⊢ ∃𝑦 𝑦 = 𝑧
4		sneq 3649	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → {𝑧} = {𝑦})
5	4	equcoms 1732	. . . . . . . . . 10 ⊢ (𝑦 = 𝑧 → {𝑧} = {𝑦})
6	3, 5	eximii 1626	. . . . . . . . 9 ⊢ ∃𝑦{𝑧} = {𝑦}
7		vex 2776	. . . . . . . . . . 11 ⊢ 𝑧 ∈ V
8	7	snex 4237	. . . . . . . . . 10 ⊢ {𝑧} ∈ V
9		eleq2 2270	. . . . . . . . . . 11 ⊢ (𝑥 = {𝑧} → (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ {𝑧}))
10		eqeq1 2213	. . . . . . . . . . . 12 ⊢ (𝑥 = {𝑧} → (𝑥 = {𝑦} ↔ {𝑧} = {𝑦}))
11	10	exbidv 1849	. . . . . . . . . . 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 2872	. . . . . . . . 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 3868	. . . . . . . 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 2203	. . . . 5 ⊢ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} = V
19	18	eleq1i 2272	. . . 4 ⊢ (∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V ↔ V ∈ V)
20	1, 19	mtbir 673	. . 3 ⊢ ¬ ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
21		uniexg 4494	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → ∪ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
22	20, 21	mto 664	. 2 ⊢ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
23	22	nelir 2475	1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V

Syntax hints:   ∧ wa 104   = wceq 1373  ∃wex 1516   ∈ wcel 2177  {cab 2192   ∉ wnel 2472  Vcvv 2773  {csn 3638  ∪ cuni 3856

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-un 4488

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-nel 2473  df-rex 2491  df-v 2775  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-uni 3857

This theorem is referenced by:  fiprc  6921