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Mirrors > Home > ILE Home > Th. List > topnex | Unicode version |
Description: The class of all topologies is a proper class. The proof uses discrete topologies and pwnex 4421. (Contributed by BJ, 2-May-2021.) |
Ref | Expression |
---|---|
topnex |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pwnex 4421 | . . . 4 | |
2 | 1 | neli 2431 | . . 3 |
3 | vex 2724 | . . . . . . . 8 | |
4 | distop 12632 | . . . . . . . 8 | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 |
6 | eleq1 2227 | . . . . . . 7 | |
7 | 5, 6 | mpbiri 167 | . . . . . 6 |
8 | 7 | exlimiv 1585 | . . . . 5 |
9 | 8 | abssi 3212 | . . . 4 |
10 | ssexg 4115 | . . . 4 | |
11 | 9, 10 | mpan 421 | . . 3 |
12 | 2, 11 | mto 652 | . 2 |
13 | 12 | nelir 2432 | 1 |
Colors of variables: wff set class |
Syntax hints: wceq 1342 wex 1479 wcel 2135 cab 2150 wnel 2429 cvv 2721 wss 3111 cpw 3553 ctop 12542 |
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 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-un 4405 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-nel 2430 df-ral 2447 df-rex 2448 df-v 2723 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-uni 3784 df-iun 3862 df-top 12543 |
This theorem is referenced by: (None) |
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