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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.
StepHypRef 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})
54equcoms 1701 . . . . . . . . . 10 (𝑦 = 𝑧 → {𝑧} = {𝑦})
63, 5eximii 1595 . . . . . . . . 9 𝑦{𝑧} = {𝑦}
7 vex 2733 . . . . . . . . . . 11 𝑧 ∈ V
87snex 4169 . . . . . . . . . 10 {𝑧} ∈ V
9 eleq2 2234 . . . . . . . . . . 11 (𝑥 = {𝑧} → (𝑧𝑥𝑧 ∈ {𝑧}))
10 eqeq1 2177 . . . . . . . . . . . 12 (𝑥 = {𝑧} → (𝑥 = {𝑦} ↔ {𝑧} = {𝑦}))
1110exbidv 1818 . . . . . . . . . . 11 (𝑥 = {𝑧} → (∃𝑦 𝑥 = {𝑦} ↔ ∃𝑦{𝑧} = {𝑦}))
129, 11anbi12d 470 . . . . . . . . . 10 (𝑥 = {𝑧} → ((𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦}) ↔ (𝑧 ∈ {𝑧} ∧ ∃𝑦{𝑧} = {𝑦})))
138, 12spcev 2825 . . . . . . . . 9 ((𝑧 ∈ {𝑧} ∧ ∃𝑦{𝑧} = {𝑦}) → ∃𝑥(𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦}))
142, 6, 13mp2an 424 . . . . . . . 8 𝑥(𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦})
15 eluniab 3806 . . . . . . . 8 (𝑧 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ↔ ∃𝑥(𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦}))
1614, 15mpbir 145 . . . . . . 7 𝑧 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}}
1716, 72th 173 . . . . . 6 (𝑧 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ↔ 𝑧 ∈ V)
1817eqriv 2167 . . . . 5 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} = V
1918eleq1i 2236 . . . 4 ( {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V ↔ V ∈ V)
201, 19mtbir 666 . . 3 ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
21 uniexg 4422 . . 3 ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
2220, 21mto 657 . 2 ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
2322nelir 2438 1 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
Colors of variables: wff set class
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
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