Theorem cmptop 23289

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 2730	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	iscmp 23282	. 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)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054   ∩ cin 3916  𝒫 cpw 4566  ∪ cuni 4874  Fincfn 8921  Topctop 22787  Compccmp 23280

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-ss 3934  df-pw 4568  df-uni 4875  df-cmp 23281

This theorem is referenced by:  imacmp  23291  cmpcld  23296  fiuncmp  23298  cmpfii  23303  bwth  23304  locfincmp  23420  kgeni  23431  kgentopon  23432  kgencmp  23439  kgencmp2  23440  cmpkgen  23445  txcmplem1  23535  txcmp  23537  qtopcmp  23602  cmphaushmeo  23694  ptcmpfi  23707  fclscmpi  23923  alexsubALTlem1  23941  ptcmplem1  23946  ptcmpg  23951  evth  24865  evth2  24866  cmppcmp  33855  ordcmp  36442  poimirlem30  37651  heibor1lem  37810  cmpfiiin  42692  kelac1  43059  kelac2  43061  stoweidlem28  46033  stoweidlem50  46055  stoweidlem53  46058  stoweidlem57  46062  stoweidlem62  46067