Theorem isconn2 23762

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

Step	Hyp	Ref	Expression
1		isconn.1	. . 3 ⊢ 𝑋 = ∪ 𝐽
2	1	isconn 23761	. 2 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
3		eqss 3928	. . . 4 ⊢ ((𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋} ↔ ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ∧ {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽))))
4		0opn 23252	. . . . . . 7 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
5		0cld 23386	. . . . . . 7 ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽))
6	4, 5	elind 4100	. . . . . 6 ⊢ (𝐽 ∈ Top → ∅ ∈ (𝐽 ∩ (Clsd‘𝐽)))
7	1	topopn 23254	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
8	1	topcld 23383	. . . . . . 7 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (Clsd‘𝐽))
9	7, 8	elind 4100	. . . . . 6 ⊢ (𝐽 ∈ Top → 𝑋 ∈ (𝐽 ∩ (Clsd‘𝐽)))
10	6, 9	prssd 4781	. . . . 5 ⊢ (𝐽 ∈ Top → {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽)))
11	10	biantrud 541	. . . 4 ⊢ (𝐽 ∈ Top → ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ↔ ((𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋} ∧ {∅, 𝑋} ⊆ (𝐽 ∩ (Clsd‘𝐽)))))
12	3, 11	bitr4id 293	. . 3 ⊢ (𝐽 ∈ Top → ((𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋} ↔ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))
13	12	pm5.32i 585	. 2 ⊢ ((𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}) ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))
14	2, 13	bitri 278	1 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) ⊆ {∅, 𝑋}))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900   ∩ cin 4092  ∅c0 4281  {cpr 4590  ∪ cuni 4870  ‘cfv 6558  Topctop 23241  Clsdccld 23364  Conncconn 23759

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pow 5331  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6512  df-fun 6560  df-fv 6566  df-top 23242  df-cld 23367  df-conn 23760

This theorem is used by:  indisconn  23766  dfconn2  23767  cnconn  23770  txconn  24038  filconn  24232  onsucconni  37403