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Mirrors > Home > ILE Home > Th. List > innei | Unicode version |
Description: The intersection of two neighborhoods of a set is also a neighborhood of the set. Generalization to subsets of Property Vii of [BourbakiTop1] p. I.3 for binary intersections. (Contributed by FL, 28-Sep-2006.) |
Ref | Expression |
---|---|
innei |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2139 | . . . . 5 | |
2 | 1 | neii1 12316 | . . . 4 |
3 | ssinss1 3305 | . . . 4 | |
4 | 2, 3 | syl 14 | . . 3 |
5 | 4 | 3adant3 1001 | . 2 |
6 | neii2 12318 | . . . . 5 | |
7 | neii2 12318 | . . . . 5 | |
8 | 6, 7 | anim12dan 589 | . . . 4 |
9 | inopn 12170 | . . . . . . . . . . 11 | |
10 | 9 | 3expa 1181 | . . . . . . . . . 10 |
11 | ssin 3298 | . . . . . . . . . . . . 13 | |
12 | 11 | biimpi 119 | . . . . . . . . . . . 12 |
13 | ss2in 3304 | . . . . . . . . . . . 12 | |
14 | 12, 13 | anim12i 336 | . . . . . . . . . . 11 |
15 | 14 | an4s 577 | . . . . . . . . . 10 |
16 | sseq2 3121 | . . . . . . . . . . . 12 | |
17 | sseq1 3120 | . . . . . . . . . . . 12 | |
18 | 16, 17 | anbi12d 464 | . . . . . . . . . . 11 |
19 | 18 | rspcev 2789 | . . . . . . . . . 10 |
20 | 10, 15, 19 | syl2an 287 | . . . . . . . . 9 |
21 | 20 | expr 372 | . . . . . . . 8 |
22 | 21 | an32s 557 | . . . . . . 7 |
23 | 22 | rexlimdva 2549 | . . . . . 6 |
24 | 23 | rexlimdva2 2552 | . . . . 5 |
25 | 24 | imp32 255 | . . . 4 |
26 | 8, 25 | syldan 280 | . . 3 |
27 | 26 | 3impb 1177 | . 2 |
28 | 1 | neiss2 12311 | . . . 4 |
29 | 1 | isnei 12313 | . . . 4 |
30 | 28, 29 | syldan 280 | . . 3 |
31 | 30 | 3adant3 1001 | . 2 |
32 | 5, 27, 31 | mpbir2and 928 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 962 wceq 1331 wcel 1480 wrex 2417 cin 3070 wss 3071 cuni 3736 cfv 5123 ctop 12164 cnei 12307 |
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 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-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 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-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 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-top 12165 df-nei 12308 |
This theorem is referenced by: (None) |
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