Theorem connclo 23475

Description: The only nonempty clopen set of a connected topology is the whole space. (Contributed by Mario Carneiro, 10-Mar-2015.)

Hypotheses

Ref	Expression
isconn.1	⊢ 𝑋 = ∪ 𝐽
connclo.1	⊢ (𝜑 → 𝐽 ∈ Conn)
connclo.2	⊢ (𝜑 → 𝐴 ∈ 𝐽)
connclo.3	⊢ (𝜑 → 𝐴 ≠ ∅)
connclo.4	⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽))

Assertion

Ref	Expression
connclo	⊢ (𝜑 → 𝐴 = 𝑋)

Proof of Theorem connclo

Step	Hyp	Ref	Expression
1		connclo.3	. . 3 ⊢ (𝜑 → 𝐴 ≠ ∅)
2	1	neneqd 2962	. 2 ⊢ (𝜑 → ¬ 𝐴 = ∅)
3		connclo.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐽)
4		connclo.4	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽))
5	3, 4	elind 4152	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)))
6		connclo.1	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Conn)
7		isconn.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
8	7	isconn 23473	. . . . . . 7 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
9	8	simprbi 501	. . . . . 6 ⊢ (𝐽 ∈ Conn → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})
10	6, 9	syl 17	. . . . 5 ⊢ (𝜑 → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})
11	5, 10	eleqtrd 2864	. . . 4 ⊢ (𝜑 → 𝐴 ∈ {∅, 𝑋})
12		elpri 4606	. . . 4 ⊢ (𝐴 ∈ {∅, 𝑋} → (𝐴 = ∅ ∨ 𝐴 = 𝑋))
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → (𝐴 = ∅ ∨ 𝐴 = 𝑋))
14	13	ord 875	. 2 ⊢ (𝜑 → (¬ 𝐴 = ∅ → 𝐴 = 𝑋))
15	2, 14	mpd 15	1 ⊢ (𝜑 → 𝐴 = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 858   = wceq 1560   ∈ wcel 2142   ≠ wne 2957   ∩ cin 3903  ∅c0 4285  {cpr 4584  ∪ cuni 4865  ‘cfv 6521  Topctop 22953  Clsdccld 23076  Conncconn 23471

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6477  df-fv 6529  df-conn 23472

This theorem is referenced by:  conndisj  23476  cnconn  23482  connsubclo  23484  t1connperf  23496  txconn  23749  connpconn  35585  cvmliftmolem2  35632  cvmlift2lem12  35664  mblfinlem1  38156