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| Mirrors > Home > MPE Home > Th. List > cmptop | Structured version Visualization version GIF version | ||
| Description: A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.) |
| Ref | Expression |
|---|---|
| cmptop | ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
| 2 | 1 | iscmp 23396 | . 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑠))) |
| 3 | 2 | simplbi 497 | 1 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ∩ cin 3950 𝒫 cpw 4600 ∪ cuni 4907 Fincfn 8985 Topctop 22899 Compccmp 23394 |
| 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-ss 3968 df-pw 4602 df-uni 4908 df-cmp 23395 |
| This theorem is referenced by: imacmp 23405 cmpcld 23410 fiuncmp 23412 cmpfii 23417 bwth 23418 locfincmp 23534 kgeni 23545 kgentopon 23546 kgencmp 23553 kgencmp2 23554 cmpkgen 23559 txcmplem1 23649 txcmp 23651 qtopcmp 23716 cmphaushmeo 23808 ptcmpfi 23821 fclscmpi 24037 alexsubALTlem1 24055 ptcmplem1 24060 ptcmpg 24065 evth 24991 evth2 24992 cmppcmp 33857 ordcmp 36448 poimirlem30 37657 heibor1lem 37816 cmpfiiin 42708 kelac1 43075 kelac2 43077 stoweidlem28 46043 stoweidlem50 46065 stoweidlem53 46068 stoweidlem57 46072 stoweidlem62 46077 |
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