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Mirrors > Home > ILE Home > Th. List > uniopn | Unicode version |
Description: The union of a subset of a topology (that is, the union of any family of open sets of a topology) is an open set. (Contributed by Stefan Allan, 27-Feb-2006.) |
Ref | Expression |
---|---|
uniopn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istopg 12093 | . . . . 5 | |
2 | 1 | ibi 175 | . . . 4 |
3 | 2 | simpld 111 | . . 3 |
4 | elpw2g 4051 | . . . . . . . 8 | |
5 | 4 | biimpar 295 | . . . . . . 7 |
6 | sseq1 3090 | . . . . . . . . 9 | |
7 | unieq 3715 | . . . . . . . . . 10 | |
8 | 7 | eleq1d 2186 | . . . . . . . . 9 |
9 | 6, 8 | imbi12d 233 | . . . . . . . 8 |
10 | 9 | spcgv 2747 | . . . . . . 7 |
11 | 5, 10 | syl 14 | . . . . . 6 |
12 | 11 | com23 78 | . . . . 5 |
13 | 12 | ex 114 | . . . 4 |
14 | 13 | pm2.43d 50 | . . 3 |
15 | 3, 14 | mpid 42 | . 2 |
16 | 15 | imp 123 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wal 1314 wceq 1316 wcel 1465 wral 2393 cin 3040 wss 3041 cpw 3480 cuni 3706 ctop 12091 |
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-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-in 3047 df-ss 3054 df-pw 3482 df-uni 3707 df-top 12092 |
This theorem is referenced by: iunopn 12096 unopn 12099 0opn 12100 topopn 12102 tgtop 12164 ntropn 12213 neipsm 12250 unimopn 12582 metrest 12602 cnopncntop 12633 |
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