Theorem comptop 11485

Description: A compact topology is a topology.

Assertion

Ref	Expression
comptop	⊢ (J ∈ Comp → J ∈ Top)

Proof of Theorem comptop

Step	Hyp	Ref	Expression
1		iscomp 11114	. 2 ⊢ (J ∈ Comp ↔ (J ∈ Top ⋀ ∀r ∈ ℘ J(∪J = ∪r → ∃s ∈ (℘r ∩ Fin)∪J = ∪s)))
2	1	pm3.26bi 320	1 ⊢ (J ∈ Comp → J ∈ Top)

Syntax hints:   → wi 3   = wceq 992   ∈ wcel 994  ∀wral 1691  ∃wrex 1692   ∩ cin 2098  ℘cpw 2458  ∪cuni 2569  Fincfn 4508  Topctop 7800  Compccomp 11112

This theorem is referenced by:  cptclsscpt 11489  alexsublem1 11496  locfincomp 11575

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-12 1004  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500

This theorem depends on definitions:  df-bi 145  df-an 223  df-ex 1017  df-sb 1209  df-clab 1506  df-cleq 1511  df-clel 1514  df-ral 1695  df-rex 1696  df-rab 1698  df-v 1858  df-in 2103  df-ss 2105  df-pw 2459  df-uni 2570  df-comp 11113

Copyright terms: Public domain