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Theorem snnex 4307
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 4000 . . . 4 ¬ V ∈ V
2 vsnid 3504 . . . . . . . . 9 𝑧 ∈ {𝑧}
3 a9ev 1643 . . . . . . . . . 10 𝑦 𝑦 = 𝑧
4 sneq 3485 . . . . . . . . . . 11 (𝑧 = 𝑦 → {𝑧} = {𝑦})
54equcoms 1652 . . . . . . . . . 10 (𝑦 = 𝑧 → {𝑧} = {𝑦})
63, 5eximii 1549 . . . . . . . . 9 𝑦{𝑧} = {𝑦}
7 vex 2644 . . . . . . . . . . 11 𝑧 ∈ V
87snex 4049 . . . . . . . . . 10 {𝑧} ∈ V
9 eleq2 2163 . . . . . . . . . . 11 (𝑥 = {𝑧} → (𝑧𝑥𝑧 ∈ {𝑧}))
10 eqeq1 2106 . . . . . . . . . . . 12 (𝑥 = {𝑧} → (𝑥 = {𝑦} ↔ {𝑧} = {𝑦}))
1110exbidv 1764 . . . . . . . . . . 11 (𝑥 = {𝑧} → (∃𝑦 𝑥 = {𝑦} ↔ ∃𝑦{𝑧} = {𝑦}))
129, 11anbi12d 460 . . . . . . . . . 10 (𝑥 = {𝑧} → ((𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦}) ↔ (𝑧 ∈ {𝑧} ∧ ∃𝑦{𝑧} = {𝑦})))
138, 12spcev 2735 . . . . . . . . 9 ((𝑧 ∈ {𝑧} ∧ ∃𝑦{𝑧} = {𝑦}) → ∃𝑥(𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦}))
142, 6, 13mp2an 420 . . . . . . . 8 𝑥(𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦})
15 eluniab 3695 . . . . . . . 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 2097 . . . . 5 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} = V
1918eleq1i 2165 . . . 4 ( {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V ↔ V ∈ V)
201, 19mtbir 637 . . 3 ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
21 uniexg 4299 . . 3 ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
2220, 21mto 629 . 2 ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
2322nelir 2365 1 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1299  wex 1436  wcel 1448  {cab 2086  wnel 2362  Vcvv 2641  {csn 3474   cuni 3683
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-un 4293
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-nel 2363  df-rex 2381  df-v 2643  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-uni 3684
This theorem is referenced by:  fiprc  6639
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