MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isconn2 Structured version   Visualization version   GIF version

Theorem isconn2 23336
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 23335 . 2 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
3 eqss 3995 . . . 4 ((𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋} ↔ ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ∧ {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽))))
4 0opn 22824 . . . . . . 7 (𝐽 ∈ Top → ∅ ∈ 𝐽)
5 0cld 22960 . . . . . . 7 (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽))
64, 5elind 4194 . . . . . 6 (𝐽 ∈ Top → ∅ ∈ (𝐽 ∩ (Clsd‘𝐽)))
71topopn 22826 . . . . . . 7 (𝐽 ∈ Top → 𝑋𝐽)
81topcld 22957 . . . . . . 7 (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
97, 8elind 4194 . . . . . 6 (𝐽 ∈ Top → 𝑋 ∈ (𝐽 ∩ (Clsd‘𝐽)))
106, 9prssd 4828 . . . . 5 (𝐽 ∈ Top → {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽)))
1110biantrud 530 . . . 4 (𝐽 ∈ Top → ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ↔ ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ∧ {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽)))))
123, 11bitr4id 289 . . 3 (𝐽 ∈ Top → ((𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋} ↔ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))
1312pm5.32i 573 . 2 ((𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}) ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))
142, 13bitri 274 1 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  wcel 2098  cin 3946  wss 3947  c0 4324  {cpr 4632   cuni 4910  cfv 6551  Topctop 22813  Clsdccld 22938  Conncconn 23333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-iota 6503  df-fun 6553  df-fv 6559  df-top 22814  df-cld 22941  df-conn 23334
This theorem is referenced by:  indisconn  23340  dfconn2  23341  cnconn  23344  txconn  23611  filconn  23805  onsucconni  35926
  Copyright terms: Public domain W3C validator