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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 4830 | . . 3 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽) | |
2 | 1open.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 1, 2 | sseqtrrdi 3966 | . 2 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ 𝑋) |
4 | 3 | adantl 485 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ∪ cuni 4800 Topctop 21498 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-v 3443 df-in 3888 df-ss 3898 df-uni 4801 |
This theorem is referenced by: riinopn 21513 opncld 21638 ntrval2 21656 ntrss3 21665 cmclsopn 21667 opncldf1 21689 opnneissb 21719 opnssneib 21720 opnneiss 21723 neitr 21785 restntr 21787 cnpnei 21869 imasnopn 22295 cnextcn 22672 utopreg 22858 ist0cld 31186 opnregcld 33791 ptrecube 35057 poimirlem29 35086 poimir 35090 |
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