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Mirrors > Home > ILE Home > Th. List > neiint | Unicode version |
Description: An intuitive definition of a neighborhood in terms of interior. (Contributed by Szymon Jaroszewicz, 18-Dec-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
neifval.1 |
Ref | Expression |
---|---|
neiint |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neifval.1 | . . . . 5 | |
2 | 1 | isnei 12691 | . . . 4 |
3 | 2 | 3adant3 1006 | . . 3 |
4 | 3 | 3anibar 1154 | . 2 |
5 | simprrl 529 | . . . . 5 | |
6 | 1 | ssntr 12669 | . . . . . . 7 |
7 | 6 | 3adantl2 1143 | . . . . . 6 |
8 | 7 | adantrrl 478 | . . . . 5 |
9 | 5, 8 | sstrd 3147 | . . . 4 |
10 | 9 | rexlimdvaa 2582 | . . 3 |
11 | simpl1 989 | . . . . . 6 | |
12 | simpl3 991 | . . . . . 6 | |
13 | 1 | ntropn 12664 | . . . . . 6 |
14 | 11, 12, 13 | syl2anc 409 | . . . . 5 |
15 | simpr 109 | . . . . 5 | |
16 | 1 | ntrss2 12668 | . . . . . 6 |
17 | 11, 12, 16 | syl2anc 409 | . . . . 5 |
18 | sseq2 3161 | . . . . . . 7 | |
19 | sseq1 3160 | . . . . . . 7 | |
20 | 18, 19 | anbi12d 465 | . . . . . 6 |
21 | 20 | rspcev 2825 | . . . . 5 |
22 | 14, 15, 17, 21 | syl12anc 1225 | . . . 4 |
23 | 22 | ex 114 | . . 3 |
24 | 10, 23 | impbid 128 | . 2 |
25 | 4, 24 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 wrex 2443 wss 3111 cuni 3783 cfv 5182 ctop 12542 cnt 12640 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-13 2137 ax-14 2138 ax-ext 2146 ax-coll 4091 ax-sep 4094 ax-pow 4147 ax-pr 4181 ax-un 4405 |
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-ntr 12643 df-nei 12686 |
This theorem is referenced by: topssnei 12709 iscnp4 12765 |
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