Theorem connclo 23393

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 2938	. 2 ⊢ (𝜑 → ¬ 𝐴 = ∅)
3		connclo.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐽)
4		connclo.4	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽))
5	3, 4	elind 4141	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)))
6		connclo.1	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Conn)
7		isconn.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
8	7	isconn 23391	. . . . . . 7 ⊢ (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
9	8	simprbi 497	. . . . . 6 ⊢ (𝐽 ∈ Conn → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})
10	6, 9	syl 17	. . . . 5 ⊢ (𝜑 → (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋})
11	5, 10	eleqtrd 2839	. . . 4 ⊢ (𝜑 → 𝐴 ∈ {∅, 𝑋})
12		elpri 4592	. . . 4 ⊢ (𝐴 ∈ {∅, 𝑋} → (𝐴 = ∅ ∨ 𝐴 = 𝑋))
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → (𝐴 = ∅ ∨ 𝐴 = 𝑋))
14	13	ord 865	. 2 ⊢ (𝜑 → (¬ 𝐴 = ∅ → 𝐴 = 𝑋))
15	2, 14	mpd 15	1 ⊢ (𝜑 → 𝐴 = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   ∩ cin 3889  ∅c0 4274  {cpr 4570  ∪ cuni 4851  ‘cfv 6493  Topctop 22871  Clsdccld 22994  Conncconn 23389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-iota 6449  df-fv 6501  df-conn 23390

This theorem is referenced by:  conndisj  23394  cnconn  23400  connsubclo  23402  t1connperf  23414  txconn  23667  connpconn  35436  cvmliftmolem2  35483  cvmlift2lem12  35515  mblfinlem1  37995