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Theorem iscon2 21027
 Description: The predicate 𝐽 is a connected topology . (Contributed by Mario Carneiro, 10-Mar-2015.)
Hypothesis
Ref Expression
iscon.1 𝑋 = 𝐽
Assertion
Ref Expression
iscon2 (𝐽 ∈ Con ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))

Proof of Theorem iscon2
StepHypRef Expression
1 iscon.1 . . 3 𝑋 = 𝐽
21iscon 21026 . 2 (𝐽 ∈ Con ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
3 0opn 20534 . . . . . . 7 (𝐽 ∈ Top → ∅ ∈ 𝐽)
4 0cld 20652 . . . . . . 7 (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽))
53, 4elind 3760 . . . . . 6 (𝐽 ∈ Top → ∅ ∈ (𝐽 ∩ (Clsd‘𝐽)))
61topopn 20536 . . . . . . 7 (𝐽 ∈ Top → 𝑋𝐽)
71topcld 20649 . . . . . . 7 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
86, 7elind 3760 . . . . . 6 (𝐽 ∈ Top → 𝑋 ∈ (𝐽 ∩ (Clsd‘𝐽)))
9 prssi 4293 . . . . . 6 ((∅ ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝑋 ∈ (𝐽 ∩ (Clsd‘𝐽))) → {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽)))
105, 8, 9syl2anc 691 . . . . 5 (𝐽 ∈ Top → {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽)))
1110biantrud 527 . . . 4 (𝐽 ∈ Top → ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ↔ ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ∧ {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽)))))
12 eqss 3583 . . . 4 ((𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋} ↔ ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ∧ {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽))))
1311, 12syl6rbbr 278 . . 3 (𝐽 ∈ Top → ((𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋} ↔ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))
1413pm5.32i 667 . 2 ((𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}) ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))
152, 14bitri 263 1 (𝐽 ∈ Con ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {cpr 4127  ∪ cuni 4372  ‘cfv 5804  Topctop 20517  Clsdccld 20630  Conccon 21024 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-top 20521  df-cld 20633  df-con 21025 This theorem is referenced by:  indiscon  21031  dfcon2  21032  cnconn  21035  txcon  21302  filcon  21497  onsucconi  31606
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