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Mirrors > Home > ILE Home > Th. List > fiinopn | Unicode version |
Description: The intersection of a nonempty finite family of open sets is open. (Contributed by FL, 20-Apr-2012.) |
Ref | Expression |
---|---|
fiinopn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elpwg 3551 | . . . . . . 7 | |
2 | sseq1 3151 | . . . . . . . . . . . . . 14 | |
3 | neeq1 2340 | . . . . . . . . . . . . . 14 | |
4 | eleq1 2220 | . . . . . . . . . . . . . 14 | |
5 | 2, 3, 4 | 3anbi123d 1294 | . . . . . . . . . . . . 13 |
6 | inteq 3810 | . . . . . . . . . . . . . . 15 | |
7 | 6 | eleq1d 2226 | . . . . . . . . . . . . . 14 |
8 | 7 | imbi2d 229 | . . . . . . . . . . . . 13 |
9 | 5, 8 | imbi12d 233 | . . . . . . . . . . . 12 |
10 | sp 1491 | . . . . . . . . . . . . . 14 | |
11 | 10 | adantl 275 | . . . . . . . . . . . . 13 |
12 | istopfin 12409 | . . . . . . . . . . . . 13 | |
13 | 11, 12 | syl11 31 | . . . . . . . . . . . 12 |
14 | 9, 13 | vtoclg 2772 | . . . . . . . . . . 11 |
15 | 14 | com12 30 | . . . . . . . . . 10 |
16 | 15 | 3exp 1184 | . . . . . . . . 9 |
17 | 16 | com3r 79 | . . . . . . . 8 |
18 | 17 | com4r 86 | . . . . . . 7 |
19 | 1, 18 | syl6bir 163 | . . . . . 6 |
20 | 19 | pm2.43a 51 | . . . . 5 |
21 | 20 | com4l 84 | . . . 4 |
22 | 21 | pm2.43i 49 | . . 3 |
23 | 22 | 3imp 1176 | . 2 |
24 | 23 | com12 30 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 963 wal 1333 wceq 1335 wcel 2128 wne 2327 wss 3102 c0 3394 cpw 3543 cuni 3772 cint 3807 cfn 6685 ctop 12406 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-br 3966 df-opab 4026 df-id 4253 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-er 6480 df-en 6686 df-fin 6688 df-top 12407 |
This theorem is referenced by: (None) |
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