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Mirrors > Home > MPE Home > Th. List > toponss | Structured version Visualization version Unicode version |
Description: A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
toponss | TopOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elssuni 4467 | . . 3 | |
2 | 1 | adantl 482 | . 2 TopOn |
3 | toponuni 20719 | . . 3 TopOn | |
4 | 3 | adantr 481 | . 2 TopOn |
5 | 2, 4 | sseqtr4d 3642 | 1 TopOn |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 wss 3574 cuni 4436 cfv 5888 TopOnctopon 20715 |
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-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
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-rab 2921 df-v 3202 df-sbc 3436 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-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-iota 5851 df-fun 5890 df-fv 5896 df-topon 20716 |
This theorem is referenced by: en2top 20789 neiptopreu 20937 iscnp3 21048 cnntr 21079 cncnp 21084 isreg2 21181 connsub 21224 iunconnlem 21230 conncompclo 21238 1stccnp 21265 kgenidm 21350 tx1cn 21412 tx2cn 21413 xkoccn 21422 txcnp 21423 ptcnplem 21424 xkoinjcn 21490 idqtop 21509 qtopss 21518 kqfvima 21533 kqsat 21534 kqreglem1 21544 kqreglem2 21545 qtopf1 21619 fbflim 21780 flimcf 21786 flimrest 21787 isflf 21797 fclscf 21829 subgntr 21910 ghmcnp 21918 qustgpopn 21923 qustgplem 21924 tsmsxplem1 21956 tsmsxp 21958 ressusp 22069 mopnss 22251 xrtgioo 22609 lebnumlem2 22761 cfilfcls 23072 iscmet3lem2 23090 dvres3a 23678 dvmptfsum 23738 dvcnvlem 23739 dvcnv 23740 efopn 24404 dvatan 24662 txomap 29901 cnllysconn 31227 cvmlift2lem9a 31285 icccncfext 40100 dvmptconst 40129 dvmptidg 40131 qndenserrnopnlem 40517 opnvonmbllem2 40847 |
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