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Mirrors > Home > ILE Home > Th. List > inopn | Unicode version |
Description: The intersection of two open sets of a topology is an open set. (Contributed by NM, 17-Jul-2006.) |
Ref | Expression |
---|---|
inopn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | istopg 12093 | . . . . 5 | |
2 | 1 | ibi 175 | . . . 4 |
3 | 2 | simprd 113 | . . 3 |
4 | ineq1 3240 | . . . . 5 | |
5 | 4 | eleq1d 2186 | . . . 4 |
6 | ineq2 3241 | . . . . 5 | |
7 | 6 | eleq1d 2186 | . . . 4 |
8 | 5, 7 | rspc2v 2776 | . . 3 |
9 | 3, 8 | syl5com 29 | . 2 |
10 | 9 | 3impib 1164 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 947 wal 1314 wceq 1316 wcel 1465 wral 2393 cin 3040 wss 3041 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-3an 949 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-v 2662 df-in 3047 df-ss 3054 df-pw 3482 df-top 12092 |
This theorem is referenced by: tgclb 12161 topbas 12163 difopn 12204 uncld 12209 ntrin 12220 innei 12259 restopnb 12277 cnptoprest 12335 txcnp 12367 txcnmpt 12369 mopnin 12583 |
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