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Theorem isconn2 21545
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 21544 . 2 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
3 0opn 21036 . . . . . . 7 (𝐽 ∈ Top → ∅ ∈ 𝐽)
4 0cld 21170 . . . . . . 7 (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽))
53, 4elind 3997 . . . . . 6 (𝐽 ∈ Top → ∅ ∈ (𝐽 ∩ (Clsd‘𝐽)))
61topopn 21038 . . . . . . 7 (𝐽 ∈ Top → 𝑋𝐽)
71topcld 21167 . . . . . . 7 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
86, 7elind 3997 . . . . . 6 (𝐽 ∈ Top → 𝑋 ∈ (𝐽 ∩ (Clsd‘𝐽)))
9 prssi 4541 . . . . . 6 ((∅ ∈ (𝐽 ∩ (Clsd‘𝐽)) ∧ 𝑋 ∈ (𝐽 ∩ (Clsd‘𝐽))) → {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽)))
105, 8, 9syl2anc 580 . . . . 5 (𝐽 ∈ Top → {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽)))
1110biantrud 528 . . . 4 (𝐽 ∈ Top → ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ↔ ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ∧ {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽)))))
12 eqss 3814 . . . 4 ((𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋} ↔ ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ∧ {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽))))
1311, 12syl6rbbr 282 . . 3 (𝐽 ∈ Top → ((𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋} ↔ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))
1413pm5.32i 571 . 2 ((𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}) ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))
152, 14bitri 267 1 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 385   = wceq 1653  wcel 2157  cin 3769  wss 3770  c0 4116  {cpr 4371   cuni 4629  cfv 6102  Topctop 21025  Clsdccld 21148  Conncconn 21542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pow 5036  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-pw 4352  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-iota 6065  df-fun 6104  df-fv 6110  df-top 21026  df-cld 21151  df-conn 21543
This theorem is referenced by:  indisconn  21549  dfconn2  21550  cnconn  21553  txconn  21820  filconn  22014  onsucconni  32943
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