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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 4660 | . . 3 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽) | |
2 | 1open.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 1, 2 | syl6sseqr 3849 | . 2 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ 𝑋) |
4 | 3 | adantl 474 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 385 = wceq 1653 ∈ wcel 2157 ⊆ wss 3770 ∪ cuni 4629 Topctop 21025 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2778 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-clab 2787 df-cleq 2793 df-clel 2796 df-nfc 2931 df-v 3388 df-in 3777 df-ss 3784 df-uni 4630 |
This theorem is referenced by: riinopn 21040 opncld 21165 ntrval2 21183 ntrss3 21192 cmclsopn 21194 opncldf1 21216 opnneissb 21246 opnssneib 21247 opnneiss 21250 neitr 21312 restntr 21314 cnpnei 21396 imasnopn 21821 cnextcn 22198 utopreg 22383 opnregcld 32836 ptrecube 33897 poimirlem29 33926 poimir 33930 |
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