Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > t0top | Structured version Visualization version GIF version |
Description: A T0 space is a topological space. (Contributed by Jeff Hankins, 1-Feb-2010.) |
Ref | Expression |
---|---|
t0top | ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2622 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
2 | 1 | ist0 21124 | . 2 ⊢ (𝐽 ∈ Kol2 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ ∪ 𝐽∀𝑦 ∈ ∪ 𝐽(∀𝑜 ∈ 𝐽 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦))) |
3 | 2 | simplbi 476 | 1 ⊢ (𝐽 ∈ Kol2 → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∈ wcel 1990 ∀wral 2912 ∪ cuni 4436 Topctop 20698 Kol2ct0 21110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-uni 4437 df-t0 21117 |
This theorem is referenced by: restt0 21170 sst0 21177 kqt0 21549 t0hmph 21593 kqhmph 21622 ordtopt0 32441 |
Copyright terms: Public domain | W3C validator |