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Mirrors > Home > MPE Home > Th. List > eltopss | Structured version Visualization version GIF version |
Description: A member of a topology is a subset of its underlying set. (Contributed by NM, 12-Sep-2006.) |
Ref | Expression |
---|---|
1open.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
eltopss | ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elssuni 4945 | . . 3 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽) | |
2 | 1open.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 1, 2 | sseqtrrdi 4031 | . 2 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ 𝑋) |
4 | 3 | adantl 480 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ⊆ wss 3947 ∪ cuni 4913 Topctop 22886 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1537 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-v 3464 df-ss 3964 df-uni 4914 |
This theorem is referenced by: riinopn 22901 opncld 23028 ntrval2 23046 ntrss3 23055 cmclsopn 23057 opncldf1 23079 opnneissb 23109 opnssneib 23110 opnneiss 23113 neitr 23175 restntr 23177 cnpnei 23259 imasnopn 23685 cnextcn 24062 utopreg 24248 ist0cld 33648 opnregcld 36042 ptrecube 37321 poimirlem29 37350 poimir 37354 seposep 48259 iscnrm3rlem7 48280 |
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