Theorem conntop 23382

Description: A connected topology is a topology. (Contributed by FL, 22-Dec-2008.) (Revised by Mario Carneiro, 14-Dec-2013.)

Assertion

Ref	Expression
conntop	⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top)

Proof of Theorem conntop

Step	Hyp	Ref	Expression
1		eqid 2737	. . 3 ⊢ ∪ 𝐽 = ∪ 𝐽
2	1	isconn 23378	. 2 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, ∪ 𝐽}))
3	2	simplbi 496	1 ⊢ (𝐽 ∈ Conn → 𝐽 ∈ Top)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∩ cin 3889  ∅c0 4274  {cpr 4570  ∪ cuni 4851  ‘cfv 6499  Topctop 22858  Clsdccld 22981  Conncconn 23376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6455  df-fv 6507  df-conn 23377

This theorem is referenced by:  conncompss  23398  txconn  23654  qtopconn  23674  ufildr  23896  connpconn  35417  cvmliftmolem1  35463  cvmliftmolem2  35464  ordtopconn  36621