Theorem connclo 22566

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 2948	. 2 ⊢ (𝜑 → ¬ 𝐴 = ∅)
3		connclo.2	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ 𝐽)
4		connclo.4	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ (Clsd‘𝐽))
5	3, 4	elind 4128	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐽 ∩ (Clsd‘𝐽)))
6		connclo.1	. . . . . 6 ⊢ (𝜑 → 𝐽 ∈ Conn)
7		isconn.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
8	7	isconn 22564	. . . . . . 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 2841	. . . 4 ⊢ (𝜑 → 𝐴 ∈ {∅, 𝑋})
12		elpri 4583	. . . 4 ⊢ (𝐴 ∈ {∅, 𝑋} → (𝐴 = ∅ ∨ 𝐴 = 𝑋))
13	11, 12	syl 17	. . 3 ⊢ (𝜑 → (𝐴 = ∅ ∨ 𝐴 = 𝑋))
14	13	ord 861	. 2 ⊢ (𝜑 → (¬ 𝐴 = ∅ → 𝐴 = 𝑋))
15	2, 14	mpd 15	1 ⊢ (𝜑 → 𝐴 = 𝑋)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 844   = wceq 1539   ∈ wcel 2106   ≠ wne 2943   ∩ cin 3886  ∅c0 4256  {cpr 4563  ∪ cuni 4839  ‘cfv 6433  Topctop 22042  Clsdccld 22167  Conncconn 22562

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441  df-conn 22563

This theorem is referenced by:  conndisj  22567  cnconn  22573  connsubclo  22575  t1connperf  22587  txconn  22840  connpconn  33197  cvmliftmolem2  33244  cvmlift2lem12  33276  mblfinlem1  35814