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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 2170 | . . . . 5 | |
2 | 1 | neii1 12941 | . . . 4 |
3 | ssinss1 3356 | . . . 4 | |
4 | 2, 3 | syl 14 | . . 3 |
5 | 4 | 3adant3 1012 | . 2 |
6 | neii2 12943 | . . . . 5 | |
7 | neii2 12943 | . . . . 5 | |
8 | 6, 7 | anim12dan 595 | . . . 4 |
9 | inopn 12795 | . . . . . . . . . . 11 | |
10 | 9 | 3expa 1198 | . . . . . . . . . 10 |
11 | ssin 3349 | . . . . . . . . . . . . 13 | |
12 | 11 | biimpi 119 | . . . . . . . . . . . 12 |
13 | ss2in 3355 | . . . . . . . . . . . 12 | |
14 | 12, 13 | anim12i 336 | . . . . . . . . . . 11 |
15 | 14 | an4s 583 | . . . . . . . . . 10 |
16 | sseq2 3171 | . . . . . . . . . . . 12 | |
17 | sseq1 3170 | . . . . . . . . . . . 12 | |
18 | 16, 17 | anbi12d 470 | . . . . . . . . . . 11 |
19 | 18 | rspcev 2834 | . . . . . . . . . 10 |
20 | 10, 15, 19 | syl2an 287 | . . . . . . . . 9 |
21 | 20 | expr 373 | . . . . . . . 8 |
22 | 21 | an32s 563 | . . . . . . 7 |
23 | 22 | rexlimdva 2587 | . . . . . 6 |
24 | 23 | rexlimdva2 2590 | . . . . 5 |
25 | 24 | imp32 255 | . . . 4 |
26 | 8, 25 | syldan 280 | . . 3 |
27 | 26 | 3impb 1194 | . 2 |
28 | 1 | neiss2 12936 | . . . 4 |
29 | 1 | isnei 12938 | . . . 4 |
30 | 28, 29 | syldan 280 | . . 3 |
31 | 30 | 3adant3 1012 | . 2 |
32 | 5, 27, 31 | mpbir2and 939 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 973 wceq 1348 wcel 2141 wrex 2449 cin 3120 wss 3121 cuni 3796 cfv 5198 ctop 12789 cnei 12932 |
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 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-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 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-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 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-top 12790 df-nei 12933 |
This theorem is referenced by: (None) |
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