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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 4068 |
. . . 4
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2 | vsnid 3564 |
. . . . . . . . 9
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3 | a9ev 1676 |
. . . . . . . . . 10
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4 | sneq 3543 |
. . . . . . . . . . 11
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5 | 4 | equcoms 1685 |
. . . . . . . . . 10
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6 | 3, 5 | eximii 1582 |
. . . . . . . . 9
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7 | vex 2692 |
. . . . . . . . . . 11
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8 | 7 | snex 4117 |
. . . . . . . . . 10
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9 | eleq2 2204 |
. . . . . . . . . . 11
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10 | eqeq1 2147 |
. . . . . . . . . . . 12
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11 | 10 | exbidv 1798 |
. . . . . . . . . . 11
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12 | 9, 11 | anbi12d 465 |
. . . . . . . . . 10
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13 | 8, 12 | spcev 2784 |
. . . . . . . . 9
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14 | 2, 6, 13 | mp2an 423 |
. . . . . . . 8
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15 | eluniab 3756 |
. . . . . . . 8
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16 | 14, 15 | mpbir 145 |
. . . . . . 7
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17 | 16, 7 | 2th 173 |
. . . . . 6
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18 | 17 | eqriv 2137 |
. . . . 5
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19 | 18 | eleq1i 2206 |
. . . 4
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20 | 1, 19 | mtbir 661 |
. . 3
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21 | uniexg 4369 |
. . 3
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22 | 20, 21 | mto 652 |
. 2
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23 | 22 | nelir 2407 |
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 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-un 4363 |
This theorem depends on definitions: df-bi 116 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-nel 2405 df-rex 2423 df-v 2691 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-uni 3745 |
This theorem is referenced by: fiprc 6717 |
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