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Mirrors > Home > MPE Home > Th. List > Mathboxes > pibp16 | Structured version Visualization version GIF version |
Description: Property P000016 of pi-base. The class of compact topologies. A space 𝑋 is compact if every open cover of 𝑋 has a finite subcover. This theorem is just a relabelled copy of iscmp 22239. (Contributed by ML, 8-Dec-2020.) |
Ref | Expression |
---|---|
pibp16.x | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
pibp16 | ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pibp16.x | . 2 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | iscmp 22239 | 1 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑦 → ∃𝑧 ∈ (𝒫 𝑦 ∩ Fin)𝑋 = ∪ 𝑧))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 ∩ cin 3852 𝒫 cpw 4499 ∪ cuni 4805 Fincfn 8604 Topctop 21744 Compccmp 22237 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1546 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-in 3860 df-ss 3870 df-pw 4501 df-uni 4806 df-cmp 22238 |
This theorem is referenced by: pibt1 35273 |
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