Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 2164 | . . . . 5 | |
2 | 1 | neii1 12688 | . . . 4 |
3 | ssinss1 3346 | . . . 4 | |
4 | 2, 3 | syl 14 | . . 3 |
5 | 4 | 3adant3 1006 | . 2 |
6 | neii2 12690 | . . . . 5 | |
7 | neii2 12690 | . . . . 5 | |
8 | 6, 7 | anim12dan 590 | . . . 4 |
9 | inopn 12542 | . . . . . . . . . . 11 | |
10 | 9 | 3expa 1192 | . . . . . . . . . 10 |
11 | ssin 3339 | . . . . . . . . . . . . 13 | |
12 | 11 | biimpi 119 | . . . . . . . . . . . 12 |
13 | ss2in 3345 | . . . . . . . . . . . 12 | |
14 | 12, 13 | anim12i 336 | . . . . . . . . . . 11 |
15 | 14 | an4s 578 | . . . . . . . . . 10 |
16 | sseq2 3161 | . . . . . . . . . . . 12 | |
17 | sseq1 3160 | . . . . . . . . . . . 12 | |
18 | 16, 17 | anbi12d 465 | . . . . . . . . . . 11 |
19 | 18 | rspcev 2825 | . . . . . . . . . 10 |
20 | 10, 15, 19 | syl2an 287 | . . . . . . . . 9 |
21 | 20 | expr 373 | . . . . . . . 8 |
22 | 21 | an32s 558 | . . . . . . 7 |
23 | 22 | rexlimdva 2581 | . . . . . 6 |
24 | 23 | rexlimdva2 2584 | . . . . 5 |
25 | 24 | imp32 255 | . . . 4 |
26 | 8, 25 | syldan 280 | . . 3 |
27 | 26 | 3impb 1188 | . 2 |
28 | 1 | neiss2 12683 | . . . 4 |
29 | 1 | isnei 12685 | . . . 4 |
30 | 28, 29 | syldan 280 | . . 3 |
31 | 30 | 3adant3 1006 | . 2 |
32 | 5, 27, 31 | mpbir2and 933 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 wrex 2443 cin 3110 wss 3111 cuni 3783 cfv 5182 ctop 12536 cnei 12679 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rab 2451 df-v 2723 df-sbc 2947 df-csb 3041 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-iun 3862 df-br 3977 df-opab 4038 df-mpt 4039 df-id 4265 df-xp 4604 df-rel 4605 df-cnv 4606 df-co 4607 df-dm 4608 df-rn 4609 df-res 4610 df-ima 4611 df-iota 5147 df-fun 5184 df-fn 5185 df-f 5186 df-f1 5187 df-fo 5188 df-f1o 5189 df-fv 5190 df-top 12537 df-nei 12680 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |