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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 3574 | . . . . . . 7 | |
2 | sseq1 3170 | . . . . . . . . . . . . . 14 | |
3 | neeq1 2353 | . . . . . . . . . . . . . 14 | |
4 | eleq1 2233 | . . . . . . . . . . . . . 14 | |
5 | 2, 3, 4 | 3anbi123d 1307 | . . . . . . . . . . . . 13 |
6 | inteq 3834 | . . . . . . . . . . . . . . 15 | |
7 | 6 | eleq1d 2239 | . . . . . . . . . . . . . 14 |
8 | 7 | imbi2d 229 | . . . . . . . . . . . . 13 |
9 | 5, 8 | imbi12d 233 | . . . . . . . . . . . 12 |
10 | sp 1504 | . . . . . . . . . . . . . 14 | |
11 | 10 | adantl 275 | . . . . . . . . . . . . 13 |
12 | istopfin 12792 | . . . . . . . . . . . . 13 | |
13 | 11, 12 | syl11 31 | . . . . . . . . . . . 12 |
14 | 9, 13 | vtoclg 2790 | . . . . . . . . . . 11 |
15 | 14 | com12 30 | . . . . . . . . . 10 |
16 | 15 | 3exp 1197 | . . . . . . . . 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 1188 | . 2 |
24 | 23 | com12 30 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 973 wal 1346 wceq 1348 wcel 2141 wne 2340 wss 3121 c0 3414 cpw 3566 cuni 3796 cint 3831 cfn 6718 ctop 12789 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-id 4278 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-er 6513 df-en 6719 df-fin 6721 df-top 12790 |
This theorem is referenced by: (None) |
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