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

Theorem conndisj 23425
Description: If a topology is connected, its underlying set can't be partitioned into two nonempty non-overlapping open sets. (Contributed by FL, 16-Nov-2008.) (Proof shortened by Mario Carneiro, 10-Mar-2015.)
Hypotheses
Ref Expression
isconn.1 𝑋 = 𝐽
connclo.1 (𝜑𝐽 ∈ Conn)
connclo.2 (𝜑𝐴𝐽)
connclo.3 (𝜑𝐴 ≠ ∅)
conndisj.4 (𝜑𝐵𝐽)
conndisj.5 (𝜑𝐵 ≠ ∅)
conndisj.6 (𝜑 → (𝐴𝐵) = ∅)
Assertion
Ref Expression
conndisj (𝜑 → (𝐴𝐵) ≠ 𝑋)

Proof of Theorem conndisj
StepHypRef Expression
1 connclo.3 . 2 (𝜑𝐴 ≠ ∅)
2 connclo.2 . . . . . . 7 (𝜑𝐴𝐽)
3 elssuni 4936 . . . . . . 7 (𝐴𝐽𝐴 𝐽)
42, 3syl 17 . . . . . 6 (𝜑𝐴 𝐽)
5 isconn.1 . . . . . 6 𝑋 = 𝐽
64, 5sseqtrrdi 4024 . . . . 5 (𝜑𝐴𝑋)
7 conndisj.6 . . . . 5 (𝜑 → (𝐴𝐵) = ∅)
8 uneqdifeq 4492 . . . . 5 ((𝐴𝑋 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) = 𝑋 ↔ (𝑋𝐴) = 𝐵))
96, 7, 8syl2anc 584 . . . 4 (𝜑 → ((𝐴𝐵) = 𝑋 ↔ (𝑋𝐴) = 𝐵))
10 simpr 484 . . . . . . 7 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋𝐴) = 𝐵)
1110difeq2d 4125 . . . . . 6 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋 ∖ (𝑋𝐴)) = (𝑋𝐵))
12 dfss4 4268 . . . . . . . 8 (𝐴𝑋 ↔ (𝑋 ∖ (𝑋𝐴)) = 𝐴)
136, 12sylib 218 . . . . . . 7 (𝜑 → (𝑋 ∖ (𝑋𝐴)) = 𝐴)
1413adantr 480 . . . . . 6 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋 ∖ (𝑋𝐴)) = 𝐴)
15 connclo.1 . . . . . . . . . 10 (𝜑𝐽 ∈ Conn)
1615adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐽 ∈ Conn)
17 conndisj.4 . . . . . . . . . 10 (𝜑𝐵𝐽)
1817adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐵𝐽)
19 conndisj.5 . . . . . . . . . 10 (𝜑𝐵 ≠ ∅)
2019adantr 480 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐵 ≠ ∅)
215isconn 23422 . . . . . . . . . . . . . 14 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
2221simplbi 497 . . . . . . . . . . . . 13 (𝐽 ∈ Conn → 𝐽 ∈ Top)
2315, 22syl 17 . . . . . . . . . . . 12 (𝜑𝐽 ∈ Top)
245opncld 23042 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐴𝐽) → (𝑋𝐴) ∈ (Clsd‘𝐽))
2523, 2, 24syl2anc 584 . . . . . . . . . . 11 (𝜑 → (𝑋𝐴) ∈ (Clsd‘𝐽))
2625adantr 480 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋𝐴) ∈ (Clsd‘𝐽))
2710, 26eqeltrrd 2841 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐵 ∈ (Clsd‘𝐽))
285, 16, 18, 20, 27connclo 23424 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐵 = 𝑋)
2928difeq2d 4125 . . . . . . 7 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋𝐵) = (𝑋𝑋))
30 difid 4375 . . . . . . 7 (𝑋𝑋) = ∅
3129, 30eqtrdi 2792 . . . . . 6 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋𝐵) = ∅)
3211, 14, 313eqtr3d 2784 . . . . 5 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐴 = ∅)
3332ex 412 . . . 4 (𝜑 → ((𝑋𝐴) = 𝐵𝐴 = ∅))
349, 33sylbid 240 . . 3 (𝜑 → ((𝐴𝐵) = 𝑋𝐴 = ∅))
3534necon3d 2960 . 2 (𝜑 → (𝐴 ≠ ∅ → (𝐴𝐵) ≠ 𝑋))
361, 35mpd 15 1 (𝜑 → (𝐴𝐵) ≠ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wne 2939  cdif 3947  cun 3948  cin 3949  wss 3950  c0 4332  {cpr 4627   cuni 4906  cfv 6560  Topctop 22900  Clsdccld 23025  Conncconn 23420
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-top 22901  df-cld 23028  df-conn 23421
This theorem is referenced by:  dfconn2  23428
  Copyright terms: Public domain W3C validator