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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 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vprc 4121 | . . . 4 | |
2 | vsnid 3615 | . . . . . . . . 9 | |
3 | a9ev 1690 | . . . . . . . . . 10 | |
4 | sneq 3594 | . . . . . . . . . . 11 | |
5 | 4 | equcoms 1701 | . . . . . . . . . 10 |
6 | 3, 5 | eximii 1595 | . . . . . . . . 9 |
7 | vex 2733 | . . . . . . . . . . 11 | |
8 | 7 | snex 4171 | . . . . . . . . . 10 |
9 | eleq2 2234 | . . . . . . . . . . 11 | |
10 | eqeq1 2177 | . . . . . . . . . . . 12 | |
11 | 10 | exbidv 1818 | . . . . . . . . . . 11 |
12 | 9, 11 | anbi12d 470 | . . . . . . . . . 10 |
13 | 8, 12 | spcev 2825 | . . . . . . . . 9 |
14 | 2, 6, 13 | mp2an 424 | . . . . . . . 8 |
15 | eluniab 3808 | . . . . . . . 8 | |
16 | 14, 15 | mpbir 145 | . . . . . . 7 |
17 | 16, 7 | 2th 173 | . . . . . 6 |
18 | 17 | eqriv 2167 | . . . . 5 |
19 | 18 | eleq1i 2236 | . . . 4 |
20 | 1, 19 | mtbir 666 | . . 3 |
21 | uniexg 4424 | . . 3 | |
22 | 20, 21 | mto 657 | . 2 |
23 | 22 | nelir 2438 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1348 wex 1485 wcel 2141 cab 2156 wnel 2435 cvv 2730 csn 3583 cuni 3796 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-nel 2436 df-rex 2454 df-v 2732 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-uni 3797 |
This theorem is referenced by: fiprc 6793 |
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