Theorem cmptop 23351

Description: A compact topology is a topology. (Contributed by Jeff Hankins, 29-Jun-2009.)

Assertion

Ref	Expression
cmptop	⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)

Proof of Theorem cmptop

Dummy variables 𝑠 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	iscmp 23344	. 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑠)))
3	2	simplbi 496	1 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∩ cin 3902  𝒫 cpw 4556  ∪ cuni 4865  Fincfn 8895  Topctop 22849  Compccmp 23342

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-ss 3920  df-pw 4558  df-uni 4866  df-cmp 23343

This theorem is referenced by:  imacmp  23353  cmpcld  23358  fiuncmp  23360  cmpfii  23365  bwth  23366  locfincmp  23482  kgeni  23493  kgentopon  23494  kgencmp  23501  kgencmp2  23502  cmpkgen  23507  txcmplem1  23597  txcmp  23599  qtopcmp  23664  cmphaushmeo  23756  ptcmpfi  23769  fclscmpi  23985  alexsubALTlem1  24003  ptcmplem1  24008  ptcmpg  24013  evth  24926  evth2  24927  cmppcmp  34035  ordcmp  36660  poimirlem30  37895  heibor1lem  38054  cmpfiiin  43048  kelac1  43414  kelac2  43416  stoweidlem28  46380  stoweidlem50  46402  stoweidlem53  46405  stoweidlem57  46409  stoweidlem62  46414