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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 3518 | . . . . . . 7 | |
2 | sseq1 3120 | . . . . . . . . . . . . . 14 | |
3 | neeq1 2321 | . . . . . . . . . . . . . 14 | |
4 | eleq1 2202 | . . . . . . . . . . . . . 14 | |
5 | 2, 3, 4 | 3anbi123d 1290 | . . . . . . . . . . . . 13 |
6 | inteq 3774 | . . . . . . . . . . . . . . 15 | |
7 | 6 | eleq1d 2208 | . . . . . . . . . . . . . 14 |
8 | 7 | imbi2d 229 | . . . . . . . . . . . . 13 |
9 | 5, 8 | imbi12d 233 | . . . . . . . . . . . 12 |
10 | sp 1488 | . . . . . . . . . . . . . 14 | |
11 | 10 | adantl 275 | . . . . . . . . . . . . 13 |
12 | istopfin 12167 | . . . . . . . . . . . . 13 | |
13 | 11, 12 | syl11 31 | . . . . . . . . . . . 12 |
14 | 9, 13 | vtoclg 2746 | . . . . . . . . . . 11 |
15 | 14 | com12 30 | . . . . . . . . . 10 |
16 | 15 | 3exp 1180 | . . . . . . . . 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 1175 | . 2 |
24 | 23 | com12 30 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wal 1329 wceq 1331 wcel 1480 wne 2308 wss 3071 c0 3363 cpw 3510 cuni 3736 cint 3771 cfn 6634 ctop 12164 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-er 6429 df-en 6635 df-fin 6637 df-top 12165 |
This theorem is referenced by: (None) |
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