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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 4937 | . . 3 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽) | |
| 2 | 1open.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
| 3 | 1, 2 | sseqtrrdi 4025 | . 2 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ 𝑋) | 
| 4 | 3 | adantl 481 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ∪ cuni 4907 Topctop 22899 | 
| 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-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-uni 4908 | 
| This theorem is referenced by: riinopn 22914 opncld 23041 ntrval2 23059 ntrss3 23068 cmclsopn 23070 opncldf1 23092 opnneissb 23122 opnssneib 23123 opnneiss 23126 neitr 23188 restntr 23190 cnpnei 23272 imasnopn 23698 cnextcn 24075 utopreg 24261 ist0cld 33832 opnregcld 36331 ptrecube 37627 poimirlem29 37656 poimir 37660 seposep 48823 iscnrm3rlem7 48843 | 
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