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Mirrors > Home > ILE Home > Th. List > snnex | Unicode version |
Description: The class of all singletons is a proper class. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.) |
Ref | Expression |
---|---|
snnex |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vprc 4135 |
. . . 4
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2 | vsnid 3624 |
. . . . . . . . 9
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3 | a9ev 1697 |
. . . . . . . . . 10
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4 | sneq 3603 |
. . . . . . . . . . 11
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5 | 4 | equcoms 1708 |
. . . . . . . . . 10
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6 | 3, 5 | eximii 1602 |
. . . . . . . . 9
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7 | vex 2740 |
. . . . . . . . . . 11
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8 | 7 | snex 4185 |
. . . . . . . . . 10
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9 | eleq2 2241 |
. . . . . . . . . . 11
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10 | eqeq1 2184 |
. . . . . . . . . . . 12
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11 | 10 | exbidv 1825 |
. . . . . . . . . . 11
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12 | 9, 11 | anbi12d 473 |
. . . . . . . . . 10
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13 | 8, 12 | spcev 2832 |
. . . . . . . . 9
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14 | 2, 6, 13 | mp2an 426 |
. . . . . . . 8
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15 | eluniab 3821 |
. . . . . . . 8
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16 | 14, 15 | mpbir 146 |
. . . . . . 7
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17 | 16, 7 | 2th 174 |
. . . . . 6
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18 | 17 | eqriv 2174 |
. . . . 5
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19 | 18 | eleq1i 2243 |
. . . 4
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20 | 1, 19 | mtbir 671 |
. . 3
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21 | uniexg 4439 |
. . 3
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22 | 20, 21 | mto 662 |
. 2
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23 | 22 | nelir 2445 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-un 4433 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-nel 2443 df-rex 2461 df-v 2739 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-uni 3810 |
This theorem is referenced by: fiprc 6814 |
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