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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 21989. (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 21989 | 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 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∀wral 3137 ∃wrex 3138 ∩ cin 3928 𝒫 cpw 4532 ∪ cuni 4831 Fincfn 8502 Topctop 21494 Compccmp 21987 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-in 3936 df-ss 3945 df-pw 4534 df-uni 4832 df-cmp 21988 |
This theorem is referenced by: pibt1 34725 |
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