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Theorem snnex 4127
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 3858 . . . 4 ¬ V V
2 vex 2534 . . . . . . . . . 10 z V
32snid 3373 . . . . . . . . 9 z {z}
4 a9ev 1565 . . . . . . . . . 10 y y = z
5 sneq 3357 . . . . . . . . . . 11 (z = y → {z} = {y})
65equcoms 1572 . . . . . . . . . 10 (y = z → {z} = {y})
74, 6eximii 1471 . . . . . . . . 9 y{z} = {y}
8 snexgOLD 3905 . . . . . . . . . . 11 (z V → {z} V)
92, 8ax-mp 7 . . . . . . . . . 10 {z} V
10 eleq2 2079 . . . . . . . . . . 11 (x = {z} → (z xz {z}))
11 eqeq1 2024 . . . . . . . . . . . 12 (x = {z} → (x = {y} ↔ {z} = {y}))
1211exbidv 1684 . . . . . . . . . . 11 (x = {z} → (y x = {y} ↔ y{z} = {y}))
1310, 12anbi12d 445 . . . . . . . . . 10 (x = {z} → ((z x y x = {y}) ↔ (z {z} y{z} = {y})))
149, 13spcev 2620 . . . . . . . . 9 ((z {z} y{z} = {y}) → x(z x y x = {y}))
153, 7, 14mp2an 404 . . . . . . . 8 x(z x y x = {y})
16 eluniab 3562 . . . . . . . 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 2015 . . . . 5 {xy x = {y}} = V
2019eleq1i 2081 . . . 4 ( {xy x = {y}} V ↔ V V)
211, 20mtbir 583 . . 3 ¬ {xy x = {y}} V
22 uniexg 4121 . . 3 ({xy x = {y}} V → {xy x = {y}} V)
2321, 22mto 575 . 2 ¬ {xy x = {y}} V
2423nelir 2274 1 {xy x = {y}} ∉ V
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1226  wex 1358   wcel 1370  {cab 2004  wnel 2183  Vcvv 2531  {csn 3346   cuni 3550
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-un 4116
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-fal 1232  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-nel 2185  df-rex 2286  df-v 2533  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-uni 3551
This theorem is referenced by: (None)
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