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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 4937 | . . 3 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽) | |
| 2 | 1 | adantl 481 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ ∪ 𝐽) |
| 3 | toponuni 22920 | . . 3 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) | |
| 4 | 3 | adantr 480 | . 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝑋 = ∪ 𝐽) |
| 5 | 2, 4 | sseqtrrd 4021 | 1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ∪ cuni 4907 ‘cfv 6561 TopOnctopon 22916 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-topon 22917 |
| This theorem is referenced by: en2top 22992 neiptopreu 23141 iscnp3 23252 cnntr 23283 cncnp 23288 isreg2 23385 connsub 23429 iunconnlem 23435 conncompclo 23443 1stccnp 23470 kgenidm 23555 tx1cn 23617 tx2cn 23618 xkoccn 23627 txcnp 23628 ptcnplem 23629 xkoinjcn 23695 idqtop 23714 qtopss 23723 kqfvima 23738 kqsat 23739 kqreglem1 23749 kqreglem2 23750 qtopf1 23824 fbflim 23984 flimcf 23990 flimrest 23991 isflf 24001 fclscf 24033 subgntr 24115 ghmcnp 24123 qustgpopn 24128 qustgplem 24129 tsmsxplem1 24161 tsmsxp 24163 ressusp 24273 mopnss 24456 xrtgioo 24828 lebnumlem2 24994 cfilfcls 25308 iscmet3lem2 25326 dvres3a 25949 dvmptfsum 26013 dvcnvlem 26014 dvcnv 26015 efopn 26700 txomap 33833 cnllysconn 35250 cvmlift2lem9a 35308 icccncfext 45902 dvmptconst 45930 dvmptidg 45932 qndenserrnopnlem 46312 opnvonmbllem2 46648 |
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