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Theorem conndisj 23440
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 4942 . . . . . . 7 (𝐴𝐽𝐴 𝐽)
42, 3syl 17 . . . . . 6 (𝜑𝐴 𝐽)
5 isconn.1 . . . . . 6 𝑋 = 𝐽
64, 5sseqtrrdi 4047 . . . . 5 (𝜑𝐴𝑋)
7 conndisj.6 . . . . 5 (𝜑 → (𝐴𝐵) = ∅)
8 uneqdifeq 4499 . . . . 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 4136 . . . . . 6 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋 ∖ (𝑋𝐴)) = (𝑋𝐵))
12 dfss4 4275 . . . . . . . 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 23437 . . . . . . . . . . . . . 14 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
2221simplbi 497 . . . . . . . . . . . . 13 (𝐽 ∈ Conn → 𝐽 ∈ Top)
2315, 22syl 17 . . . . . . . . . . . 12 (𝜑𝐽 ∈ Top)
245opncld 23057 . . . . . . . . . . . 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 2840 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐵 ∈ (Clsd‘𝐽))
285, 16, 18, 20, 27connclo 23439 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐵 = 𝑋)
2928difeq2d 4136 . . . . . . 7 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋𝐵) = (𝑋𝑋))
30 difid 4382 . . . . . . 7 (𝑋𝑋) = ∅
3129, 30eqtrdi 2791 . . . . . 6 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋𝐵) = ∅)
3211, 14, 313eqtr3d 2783 . . . . 5 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐴 = ∅)
3332ex 412 . . . 4 (𝜑 → ((𝑋𝐴) = 𝐵𝐴 = ∅))
349, 33sylbid 240 . . 3 (𝜑 → ((𝐴𝐵) = 𝑋𝐴 = ∅))
3534necon3d 2959 . 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 1537  wcel 2106  wne 2938  cdif 3960  cun 3961  cin 3962  wss 3963  c0 4339  {cpr 4633   cuni 4912  cfv 6563  Topctop 22915  Clsdccld 23040  Conncconn 23435
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-top 22916  df-cld 23043  df-conn 23436
This theorem is referenced by:  dfconn2  23443
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