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Theorem isconn2 21265
Description: The predicate 𝐽 is a connected topology . (Contributed by Mario Carneiro, 10-Mar-2015.)
Hypothesis
Ref Expression
isconn.1 𝑋 = 𝐽
Assertion
Ref Expression
isconn2 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))

Proof of Theorem isconn2
StepHypRef Expression
1 isconn.1 . . 3 𝑋 = 𝐽
21isconn 21264 . 2 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
3 0opn 20757 . . . . . . 7 (𝐽 ∈ Top → ∅ ∈ 𝐽)
4 0cld 20890 . . . . . . 7 (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽))
53, 4elind 3831 . . . . . 6 (𝐽 ∈ Top → ∅ ∈ (𝐽 ∩ (Clsd‘𝐽)))
61topopn 20759 . . . . . . 7 (𝐽 ∈ Top → 𝑋𝐽)
71topcld 20887 . . . . . . 7 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
86, 7elind 3831 . . . . . 6 (𝐽 ∈ Top → 𝑋 ∈ (𝐽 ∩ (Clsd‘𝐽)))
9 prssi 4385 . . . . . 6 ((∅ ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝑋 ∈ (𝐽 ∩ (Clsd‘𝐽))) → {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽)))
105, 8, 9syl2anc 694 . . . . 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 3651 . . . 4 ((𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋} ↔ ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ∧ {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽))))
1311, 12syl6rbbr 279 . . 3 (𝐽 ∈ Top → ((𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋} ↔ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))
1413pm5.32i 670 . 2 ((𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}) ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))
152, 14bitri 264 1 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wcel 2030  cin 3606  wss 3607  c0 3948  {cpr 4212   cuni 4468  cfv 5926  Topctop 20746  Clsdccld 20868  Conncconn 21262
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  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  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-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-top 20747  df-cld 20871  df-conn 21263
This theorem is referenced by:  indisconn  21269  dfconn2  21270  cnconn  21273  txconn  21540  filconn  21734  onsucconni  32561
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