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Theorem snnex 4209
 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 3916 . . . 4 ¬ V ∈ V
2 vex 2577 . . . . . . . . . 10 𝑧 ∈ V
32snid 3430 . . . . . . . . 9 𝑧 ∈ {𝑧}
4 a9ev 1603 . . . . . . . . . 10 𝑦 𝑦 = 𝑧
5 sneq 3414 . . . . . . . . . . 11 (𝑧 = 𝑦 → {𝑧} = {𝑦})
65equcoms 1610 . . . . . . . . . 10 (𝑦 = 𝑧 → {𝑧} = {𝑦})
74, 6eximii 1509 . . . . . . . . 9 𝑦{𝑧} = {𝑦}
8 snexgOLD 3963 . . . . . . . . . . 11 (𝑧 ∈ V → {𝑧} ∈ V)
92, 8ax-mp 7 . . . . . . . . . 10 {𝑧} ∈ V
10 eleq2 2117 . . . . . . . . . . 11 (𝑥 = {𝑧} → (𝑧𝑥𝑧 ∈ {𝑧}))
11 eqeq1 2062 . . . . . . . . . . . 12 (𝑥 = {𝑧} → (𝑥 = {𝑦} ↔ {𝑧} = {𝑦}))
1211exbidv 1722 . . . . . . . . . . 11 (𝑥 = {𝑧} → (∃𝑦 𝑥 = {𝑦} ↔ ∃𝑦{𝑧} = {𝑦}))
1310, 12anbi12d 450 . . . . . . . . . 10 (𝑥 = {𝑧} → ((𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦}) ↔ (𝑧 ∈ {𝑧} ∧ ∃𝑦{𝑧} = {𝑦})))
149, 13spcev 2664 . . . . . . . . 9 ((𝑧 ∈ {𝑧} ∧ ∃𝑦{𝑧} = {𝑦}) → ∃𝑥(𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦}))
153, 7, 14mp2an 410 . . . . . . . 8 𝑥(𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦})
16 eluniab 3620 . . . . . . . 8 (𝑧 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ↔ ∃𝑥(𝑧𝑥 ∧ ∃𝑦 𝑥 = {𝑦}))
1715, 16mpbir 138 . . . . . . 7 𝑧 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}}
1817, 22th 167 . . . . . 6 (𝑧 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ↔ 𝑧 ∈ V)
1918eqriv 2053 . . . . 5 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} = V
2019eleq1i 2119 . . . 4 ( {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V ↔ V ∈ V)
211, 20mtbir 606 . . 3 ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
22 uniexg 4203 . . 3 ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
2321, 22mto 598 . 2 ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V
2423nelir 2317 1 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
 Colors of variables: wff set class Syntax hints:   ∧ wa 101   = wceq 1259  ∃wex 1397   ∈ wcel 1409  {cab 2042   ∉ wnel 2314  Vcvv 2574  {csn 3403  ∪ cuni 3608 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-un 4198 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-nel 2315  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-uni 3609 This theorem is referenced by:  fiprc  6323
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