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Theorem snnex 4147
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 {xy 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 3879 . . . 4 ¬ V V
2 vex 2554 . . . . . . . . . 10 z V
32snid 3394 . . . . . . . . 9 z {z}
4 a9ev 1584 . . . . . . . . . 10 y y = z
5 sneq 3378 . . . . . . . . . . 11 (z = y → {z} = {y})
65equcoms 1591 . . . . . . . . . 10 (y = z → {z} = {y})
74, 6eximii 1490 . . . . . . . . 9 y{z} = {y}
8 snexgOLD 3926 . . . . . . . . . . 11 (z V → {z} V)
92, 8ax-mp 7 . . . . . . . . . 10 {z} V
10 eleq2 2098 . . . . . . . . . . 11 (x = {z} → (z xz {z}))
11 eqeq1 2043 . . . . . . . . . . . 12 (x = {z} → (x = {y} ↔ {z} = {y}))
1211exbidv 1703 . . . . . . . . . . 11 (x = {z} → (y x = {y} ↔ y{z} = {y}))
1310, 12anbi12d 442 . . . . . . . . . 10 (x = {z} → ((z x y x = {y}) ↔ (z {z} y{z} = {y})))
149, 13spcev 2641 . . . . . . . . 9 ((z {z} y{z} = {y}) → x(z x y x = {y}))
153, 7, 14mp2an 402 . . . . . . . 8 x(z x y x = {y})
16 eluniab 3583 . . . . . . . 8 (z {xy x = {y}} ↔ x(z x y x = {y}))
1715, 16mpbir 134 . . . . . . 7 z {xy x = {y}}
1817, 22th 163 . . . . . 6 (z {xy x = {y}} ↔ z V)
1918eqriv 2034 . . . . 5 {xy x = {y}} = V
2019eleq1i 2100 . . . 4 ( {xy x = {y}} V ↔ V V)
211, 20mtbir 595 . . 3 ¬ {xy x = {y}} V
22 uniexg 4141 . . 3 ({xy x = {y}} V → {xy x = {y}} V)
2321, 22mto 587 . 2 ¬ {xy x = {y}} V
2423nelir 2294 1 {xy x = {y}} ∉ V
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  {cab 2023  wnel 2202  Vcvv 2551  {csn 3367   cuni 3571
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-nel 2204  df-rex 2306  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-uni 3572
This theorem is referenced by:  fiprc  6228
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