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Theorem snnex 4339
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 4030 . . . 4 ¬ V ∈ V
2 vsnid 3527 . . . . . . . . 9 𝑧 ∈ {𝑧}
3 a9ev 1660 . . . . . . . . . 10 𝑦 𝑦 = 𝑧
4 sneq 3508 . . . . . . . . . . 11 (𝑧 = 𝑦 → {𝑧} = {𝑦})
54equcoms 1669 . . . . . . . . . 10 (𝑦 = 𝑧 → {𝑧} = {𝑦})
63, 5eximii 1566 . . . . . . . . 9 𝑦{𝑧} = {𝑦}
7 vex 2663 . . . . . . . . . . 11 𝑧 ∈ V
87snex 4079 . . . . . . . . . 10 {𝑧} ∈ V
9 eleq2 2181 . . . . . . . . . . 11 (𝑥 = {𝑧} → (𝑧𝑥𝑧 ∈ {𝑧}))
10 eqeq1 2124 . . . . . . . . . . . 12 (𝑥 = {𝑧} → (𝑥 = {𝑦} ↔ {𝑧} = {𝑦}))
1110exbidv 1781 . . . . . . . . . . 11 (𝑥 = {𝑧} → (∃𝑦 𝑥 = {𝑦} ↔ ∃𝑦{𝑧} = {𝑦}))
129, 11anbi12d 464 . . . . . . . . . 10 (𝑥 = {𝑧} → ((𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦}) ↔ (𝑧 ∈ {𝑧} ∧ ∃𝑦{𝑧} = {𝑦})))
138, 12spcev 2754 . . . . . . . . 9 ((𝑧 ∈ {𝑧} ∧ ∃𝑦{𝑧} = {𝑦}) → ∃𝑥(𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦}))
142, 6, 13mp2an 422 . . . . . . . 8 𝑥(𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦})
15 eluniab 3718 . . . . . . . 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 2114 . . . . 5 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} = V
1918eleq1i 2183 . . . 4 ( {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V ↔ V ∈ V)
201, 19mtbir 645 . . 3 ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
21 uniexg 4331 . . 3 ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
2220, 21mto 636 . 2 ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
2322nelir 2383 1 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
Colors of variables: wff set class
Syntax hints:  wa 103   = wceq 1316  wex 1453  wcel 1465  {cab 2103  wnel 2380  Vcvv 2660  {csn 3497   cuni 3706
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-nel 2381  df-rex 2399  df-v 2662  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-uni 3707
This theorem is referenced by:  fiprc  6677
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