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| Mirrors > Home > MPE Home > Th. List > toponss | Structured version Visualization version GIF version | ||
| Description: A member of a topology is a subset of its underlying set. (Contributed by Mario Carneiro, 21-Aug-2015.) |
| Ref | Expression |
|---|---|
| toponss | β’ ((π½ β (TopOnβπ) β§ π΄ β π½) β π΄ β π) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elssuni 4942 | . . 3 β’ (π΄ β π½ β π΄ β βͺ π½) | |
| 2 | 1 | adantl 483 | . 2 β’ ((π½ β (TopOnβπ) β§ π΄ β π½) β π΄ β βͺ π½) |
| 3 | toponuni 22416 | . . 3 β’ (π½ β (TopOnβπ) β π = βͺ π½) | |
| 4 | 3 | adantr 482 | . 2 β’ ((π½ β (TopOnβπ) β§ π΄ β π½) β π = βͺ π½) |
| 5 | 2, 4 | sseqtrrd 4024 | 1 β’ ((π½ β (TopOnβπ) β§ π΄ β π½) β π΄ β π) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wss 3949 βͺ cuni 4909 βcfv 6544 TopOnctopon 22412 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
| This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-iota 6496 df-fun 6546 df-fv 6552 df-topon 22413 |
| This theorem is referenced by: en2top 22488 neiptopreu 22637 iscnp3 22748 cnntr 22779 cncnp 22784 isreg2 22881 connsub 22925 iunconnlem 22931 conncompclo 22939 1stccnp 22966 kgenidm 23051 tx1cn 23113 tx2cn 23114 xkoccn 23123 txcnp 23124 ptcnplem 23125 xkoinjcn 23191 idqtop 23210 qtopss 23219 kqfvima 23234 kqsat 23235 kqreglem1 23245 kqreglem2 23246 qtopf1 23320 fbflim 23480 flimcf 23486 flimrest 23487 isflf 23497 fclscf 23529 subgntr 23611 ghmcnp 23619 qustgpopn 23624 qustgplem 23625 tsmsxplem1 23657 tsmsxp 23659 ressusp 23769 mopnss 23952 xrtgioo 24322 lebnumlem2 24478 cfilfcls 24791 iscmet3lem2 24809 dvres3a 25431 dvmptfsum 25492 dvcnvlem 25493 dvcnv 25494 efopn 26166 txomap 32845 cnllysconn 34267 cvmlift2lem9a 34325 icccncfext 44651 dvmptconst 44679 dvmptidg 44681 qndenserrnopnlem 45061 opnvonmbllem2 45397 |
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