Theorem cmptop 23435

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 2761	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	iscmp 23428	. 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 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085   ∩ cin 3903  𝒫 cpw 4554  ∪ cuni 4864  Fincfn 8923  Topctop 22933  Compccmp 23426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-ss 3921  df-pw 4556  df-uni 4865  df-cmp 23427

This theorem is referenced by:  imacmp  23437  cmpcld  23442  fiuncmp  23444  cmpfii  23449  bwth  23450  locfincmp  23566  kgeni  23577  kgentopon  23578  kgencmp  23585  kgencmp2  23586  cmpkgen  23591  txcmplem1  23681  txcmp  23683  qtopcmp  23748  cmphaushmeo  23840  ptcmpfi  23853  fclscmpi  24069  alexsubALTlem1  24087  ptcmplem1  24092  ptcmpg  24097  evth  25001  evth2  25002  cmppcmp  34116  ordcmp  36771  poimirlem30  38113  heibor1lem  38272  cmpfiiin  43242  kelac1  43604  kelac2  43606  stoweidlem28  46566  stoweidlem50  46588  stoweidlem53  46591  stoweidlem57  46595  stoweidlem62  46600