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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 | ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elssuni 4942 | . . 3 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽) | |
2 | 1 | adantl 481 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ ∪ 𝐽) |
3 | toponuni 22936 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
4 | 3 | adantr 480 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝑋 = ∪ 𝐽) |
5 | 2, 4 | sseqtrrd 4037 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ∪ cuni 4912 ‘cfv 6563 TopOnctopon 22932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-topon 22933 |
This theorem is referenced by: en2top 23008 neiptopreu 23157 iscnp3 23268 cnntr 23299 cncnp 23304 isreg2 23401 connsub 23445 iunconnlem 23451 conncompclo 23459 1stccnp 23486 kgenidm 23571 tx1cn 23633 tx2cn 23634 xkoccn 23643 txcnp 23644 ptcnplem 23645 xkoinjcn 23711 idqtop 23730 qtopss 23739 kqfvima 23754 kqsat 23755 kqreglem1 23765 kqreglem2 23766 qtopf1 23840 fbflim 24000 flimcf 24006 flimrest 24007 isflf 24017 fclscf 24049 subgntr 24131 ghmcnp 24139 qustgpopn 24144 qustgplem 24145 tsmsxplem1 24177 tsmsxp 24179 ressusp 24289 mopnss 24472 xrtgioo 24842 lebnumlem2 25008 cfilfcls 25322 iscmet3lem2 25340 dvres3a 25964 dvmptfsum 26028 dvcnvlem 26029 dvcnv 26030 efopn 26715 txomap 33795 cnllysconn 35230 cvmlift2lem9a 35288 icccncfext 45843 dvmptconst 45871 dvmptidg 45873 qndenserrnopnlem 46253 opnvonmbllem2 46589 |
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