Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > neipsm | Unicode version |
Description: A neighborhood of a set is a neighborhood of every point in the set. Proposition 1 of [BourbakiTop1] p. I.2. (Contributed by FL, 16-Nov-2006.) (Revised by Jim Kingdon, 22-Mar-2023.) |
Ref | Expression |
---|---|
neips.1 |
Ref | Expression |
---|---|
neipsm |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 3711 | . . . . . 6 | |
2 | neiss 12697 | . . . . . 6 | |
3 | 1, 2 | syl3an3 1262 | . . . . 5 |
4 | 3 | 3exp 1191 | . . . 4 |
5 | 4 | ralrimdv 2543 | . . 3 |
6 | 5 | 3ad2ant1 1007 | . 2 |
7 | eleq1w 2225 | . . . . . . 7 | |
8 | 7 | cbvexv 1905 | . . . . . 6 |
9 | r19.28mv 3496 | . . . . . 6 | |
10 | 8, 9 | sylbir 134 | . . . . 5 |
11 | 10 | 3ad2ant3 1009 | . . . 4 |
12 | ssrab2 3222 | . . . . . . . . . 10 | |
13 | uniopn 12546 | . . . . . . . . . 10 | |
14 | 12, 13 | mpan2 422 | . . . . . . . . 9 |
15 | 14 | ad2antrr 480 | . . . . . . . 8 |
16 | sseq1 3160 | . . . . . . . . . . . . . . . 16 | |
17 | 16 | elrab 2877 | . . . . . . . . . . . . . . 15 |
18 | elunii 3788 | . . . . . . . . . . . . . . 15 | |
19 | 17, 18 | sylan2br 286 | . . . . . . . . . . . . . 14 |
20 | 19 | an12s 555 | . . . . . . . . . . . . 13 |
21 | 20 | rexlimiva 2576 | . . . . . . . . . . . 12 |
22 | 21 | ralimi 2527 | . . . . . . . . . . 11 |
23 | dfss3 3127 | . . . . . . . . . . 11 | |
24 | 22, 23 | sylibr 133 | . . . . . . . . . 10 |
25 | 24 | adantl 275 | . . . . . . . . 9 |
26 | unissb 3813 | . . . . . . . . . 10 | |
27 | sseq1 3160 | . . . . . . . . . . . 12 | |
28 | 27 | elrab 2877 | . . . . . . . . . . 11 |
29 | 28 | simprbi 273 | . . . . . . . . . 10 |
30 | 26, 29 | mprgbir 2522 | . . . . . . . . 9 |
31 | 25, 30 | jctir 311 | . . . . . . . 8 |
32 | sseq2 3161 | . . . . . . . . . 10 | |
33 | sseq1 3160 | . . . . . . . . . 10 | |
34 | 32, 33 | anbi12d 465 | . . . . . . . . 9 |
35 | 34 | rspcev 2825 | . . . . . . . 8 |
36 | 15, 31, 35 | syl2anc 409 | . . . . . . 7 |
37 | 36 | ex 114 | . . . . . 6 |
38 | 37 | anim2d 335 | . . . . 5 |
39 | 38 | 3adant3 1006 | . . . 4 |
40 | 11, 39 | sylbid 149 | . . 3 |
41 | ssel2 3132 | . . . . . . 7 | |
42 | neips.1 | . . . . . . . 8 | |
43 | 42 | isneip 12693 | . . . . . . 7 |
44 | 41, 43 | sylan2 284 | . . . . . 6 |
45 | 44 | anassrs 398 | . . . . 5 |
46 | 45 | ralbidva 2460 | . . . 4 |
47 | 46 | 3adant3 1006 | . . 3 |
48 | 42 | isnei 12691 | . . . 4 |
49 | 48 | 3adant3 1006 | . . 3 |
50 | 40, 47, 49 | 3imtr4d 202 | . 2 |
51 | 6, 50 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wex 1479 wcel 2135 wral 2442 wrex 2443 crab 2446 wss 3111 csn 3570 cuni 3783 cfv 5182 ctop 12542 cnei 12685 |
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 12543 df-nei 12686 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |