Theorem cmptop 21008

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 2610	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	iscmp 21001	. 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑠)))
3	2	simplbi 475	1 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539  𝒫 cpw 4108  ∪ cuni 4372  Fincfn 7841  Topctop 20517  Compccmp 20999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110  df-uni 4373  df-cmp 21000

This theorem is referenced by:  imacmp  21010  cmpcld  21015  fiuncmp  21017  cmpfii  21022  bwth  21023  locfincmp  21139  kgeni  21150  kgentopon  21151  kgencmp  21158  kgencmp2  21159  cmpkgen  21164  txcmplem1  21254  txcmp  21256  qtopcmp  21321  cmphaushmeo  21413  ptcmpfi  21426  fclscmpi  21643  alexsubALTlem1  21661  ptcmplem1  21666  ptcmpg  21671  evth  22566  evth2  22567  cmppcmp  29253  ordcmp  31616  poimirlem30  32609  heibor1lem  32778  cmpfiiin  36278  kelac1  36651  kelac2  36653  stoweidlem28  38921  stoweidlem50  38943  stoweidlem53  38946  stoweidlem57  38950  stoweidlem59  38952  stoweidlem62  38955