Theorem cmptop 21246

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 2651	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	iscmp 21239	. 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 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942   ∩ cin 3606  𝒫 cpw 4191  ∪ cuni 4468  Fincfn 7997  Topctop 20746  Compccmp 21237

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-in 3614  df-ss 3621  df-pw 4193  df-uni 4469  df-cmp 21238

This theorem is referenced by:  imacmp  21248  cmpcld  21253  fiuncmp  21255  cmpfii  21260  bwth  21261  locfincmp  21377  kgeni  21388  kgentopon  21389  kgencmp  21396  kgencmp2  21397  cmpkgen  21402  txcmplem1  21492  txcmp  21494  qtopcmp  21559  cmphaushmeo  21651  ptcmpfi  21664  fclscmpi  21880  alexsubALTlem1  21898  ptcmplem1  21903  ptcmpg  21908  evth  22805  evth2  22806  cmppcmp  30053  ordcmp  32571  poimirlem30  33569  heibor1lem  33738  cmpfiiin  37577  kelac1  37950  kelac2  37952  stoweidlem28  40563  stoweidlem50  40585  stoweidlem53  40588  stoweidlem57  40592  stoweidlem62  40597