Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > eltopss | 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 3814 | . . 3 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ ∪ 𝐽) | |
2 | 1open.1 | . . 3 ⊢ 𝑋 = ∪ 𝐽 | |
3 | 1, 2 | sseqtrrdi 3189 | . 2 ⊢ (𝐴 ∈ 𝐽 → 𝐴 ⊆ 𝑋) |
4 | 3 | adantl 275 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝐴 ∈ 𝐽) → 𝐴 ⊆ 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1342 ∈ wcel 2135 ⊆ wss 3114 ∪ cuni 3786 Topctop 12593 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2726 df-in 3120 df-ss 3127 df-uni 3787 |
This theorem is referenced by: ntrss3 12721 opnneissb 12753 opnssneib 12754 opnneiss 12756 cnpnei 12817 imasnopn 12897 |
Copyright terms: Public domain | W3C validator |