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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 3664 | . . . . . 6 | |
2 | neiss 12319 | . . . . . 6 | |
3 | 1, 2 | syl3an3 1251 | . . . . 5 |
4 | 3 | 3exp 1180 | . . . 4 |
5 | 4 | ralrimdv 2511 | . . 3 |
6 | 5 | 3ad2ant1 1002 | . 2 |
7 | eleq1w 2200 | . . . . . . 7 | |
8 | 7 | cbvexv 1890 | . . . . . 6 |
9 | r19.28mv 3455 | . . . . . 6 | |
10 | 8, 9 | sylbir 134 | . . . . 5 |
11 | 10 | 3ad2ant3 1004 | . . . 4 |
12 | ssrab2 3182 | . . . . . . . . . 10 | |
13 | uniopn 12168 | . . . . . . . . . 10 | |
14 | 12, 13 | mpan2 421 | . . . . . . . . 9 |
15 | 14 | ad2antrr 479 | . . . . . . . 8 |
16 | sseq1 3120 | . . . . . . . . . . . . . . . 16 | |
17 | 16 | elrab 2840 | . . . . . . . . . . . . . . 15 |
18 | elunii 3741 | . . . . . . . . . . . . . . 15 | |
19 | 17, 18 | sylan2br 286 | . . . . . . . . . . . . . 14 |
20 | 19 | an12s 554 | . . . . . . . . . . . . 13 |
21 | 20 | rexlimiva 2544 | . . . . . . . . . . . 12 |
22 | 21 | ralimi 2495 | . . . . . . . . . . 11 |
23 | dfss3 3087 | . . . . . . . . . . 11 | |
24 | 22, 23 | sylibr 133 | . . . . . . . . . 10 |
25 | 24 | adantl 275 | . . . . . . . . 9 |
26 | unissb 3766 | . . . . . . . . . 10 | |
27 | sseq1 3120 | . . . . . . . . . . . 12 | |
28 | 27 | elrab 2840 | . . . . . . . . . . 11 |
29 | 28 | simprbi 273 | . . . . . . . . . 10 |
30 | 26, 29 | mprgbir 2490 | . . . . . . . . 9 |
31 | 25, 30 | jctir 311 | . . . . . . . 8 |
32 | sseq2 3121 | . . . . . . . . . 10 | |
33 | sseq1 3120 | . . . . . . . . . 10 | |
34 | 32, 33 | anbi12d 464 | . . . . . . . . 9 |
35 | 34 | rspcev 2789 | . . . . . . . 8 |
36 | 15, 31, 35 | syl2anc 408 | . . . . . . 7 |
37 | 36 | ex 114 | . . . . . 6 |
38 | 37 | anim2d 335 | . . . . 5 |
39 | 38 | 3adant3 1001 | . . . 4 |
40 | 11, 39 | sylbid 149 | . . 3 |
41 | ssel2 3092 | . . . . . . 7 | |
42 | neips.1 | . . . . . . . 8 | |
43 | 42 | isneip 12315 | . . . . . . 7 |
44 | 41, 43 | sylan2 284 | . . . . . 6 |
45 | 44 | anassrs 397 | . . . . 5 |
46 | 45 | ralbidva 2433 | . . . 4 |
47 | 46 | 3adant3 1001 | . . 3 |
48 | 42 | isnei 12313 | . . . 4 |
49 | 48 | 3adant3 1001 | . . 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 962 wceq 1331 wex 1468 wcel 1480 wral 2416 wrex 2417 crab 2420 wss 3071 csn 3527 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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