Theorem cmptop 21997

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 2821	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	iscmp 21990	. 2 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑟 ∈ 𝒫 𝐽(∪ 𝐽 = ∪ 𝑟 → ∃𝑠 ∈ (𝒫 𝑟 ∩ Fin)∪ 𝐽 = ∪ 𝑠)))
3	2	simplbi 500	1 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139   ∩ cin 3934  𝒫 cpw 4538  ∪ cuni 4831  Fincfn 8503  Topctop 21495  Compccmp 21988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-in 3942  df-ss 3951  df-pw 4540  df-uni 4832  df-cmp 21989

This theorem is referenced by:  imacmp  21999  cmpcld  22004  fiuncmp  22006  cmpfii  22011  bwth  22012  locfincmp  22128  kgeni  22139  kgentopon  22140  kgencmp  22147  kgencmp2  22148  cmpkgen  22153  txcmplem1  22243  txcmp  22245  qtopcmp  22310  cmphaushmeo  22402  ptcmpfi  22415  fclscmpi  22631  alexsubALTlem1  22649  ptcmplem1  22654  ptcmpg  22659  evth  23557  evth2  23558  cmppcmp  31117  ordcmp  33790  poimirlem30  34916  heibor1lem  35081  cmpfiiin  39287  kelac1  39656  kelac2  39658  stoweidlem28  42307  stoweidlem50  42329  stoweidlem53  42332  stoweidlem57  42336  stoweidlem62  42341