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Mirrors > Home > MPE Home > Th. List > opnneip | Structured version Visualization version Unicode version |
Description: An open set is a neighborhood of any of its members. (Contributed by NM, 8-Mar-2007.) |
Ref | Expression |
---|---|
opnneip |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snssi 4339 | . 2 | |
2 | opnneiss 20922 | . 2 | |
3 | 1, 2 | syl3an3 1361 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 w3a 1037 wcel 1990 wss 3574 csn 4177 cfv 5888 ctop 20698 cnei 20901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-top 20699 df-nei 20902 |
This theorem is referenced by: opnnei 20924 neindisj2 20927 iscnp4 21067 cnpnei 21068 hausnei2 21157 llynlly 21280 nllyrest 21289 nllyidm 21292 hausllycmp 21297 cldllycmp 21298 txnlly 21440 flimfil 21773 flimopn 21779 fbflim2 21781 hausflimlem 21783 flimcf 21786 flimsncls 21790 fclsnei 21823 fcfnei 21839 cnextcn 21871 utopreg 22056 blnei 22307 cnllycmp 22755 flimcfil 23112 limcflf 23645 rrhre 30065 cvmlift2lem12 31296 |
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