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Theorem conndisj 22790
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 4902 . . . . . . 7 (𝐴𝐽𝐴 𝐽)
42, 3syl 17 . . . . . 6 (𝜑𝐴 𝐽)
5 isconn.1 . . . . . 6 𝑋 = 𝐽
64, 5sseqtrrdi 3999 . . . . 5 (𝜑𝐴𝑋)
7 conndisj.6 . . . . 5 (𝜑 → (𝐴𝐵) = ∅)
8 uneqdifeq 4454 . . . . 5 ((𝐴𝑋 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) = 𝑋 ↔ (𝑋𝐴) = 𝐵))
96, 7, 8syl2anc 585 . . . 4 (𝜑 → ((𝐴𝐵) = 𝑋 ↔ (𝑋𝐴) = 𝐵))
10 simpr 486 . . . . . . 7 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋𝐴) = 𝐵)
1110difeq2d 4086 . . . . . 6 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋 ∖ (𝑋𝐴)) = (𝑋𝐵))
12 dfss4 4222 . . . . . . . 8 (𝐴𝑋 ↔ (𝑋 ∖ (𝑋𝐴)) = 𝐴)
136, 12sylib 217 . . . . . . 7 (𝜑 → (𝑋 ∖ (𝑋𝐴)) = 𝐴)
1413adantr 482 . . . . . 6 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋 ∖ (𝑋𝐴)) = 𝐴)
15 connclo.1 . . . . . . . . . 10 (𝜑𝐽 ∈ Conn)
1615adantr 482 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐽 ∈ Conn)
17 conndisj.4 . . . . . . . . . 10 (𝜑𝐵𝐽)
1817adantr 482 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐵𝐽)
19 conndisj.5 . . . . . . . . . 10 (𝜑𝐵 ≠ ∅)
2019adantr 482 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐵 ≠ ∅)
215isconn 22787 . . . . . . . . . . . . . 14 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
2221simplbi 499 . . . . . . . . . . . . 13 (𝐽 ∈ Conn → 𝐽 ∈ Top)
2315, 22syl 17 . . . . . . . . . . . 12 (𝜑𝐽 ∈ Top)
245opncld 22407 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐴𝐽) → (𝑋𝐴) ∈ (Clsd‘𝐽))
2523, 2, 24syl2anc 585 . . . . . . . . . . 11 (𝜑 → (𝑋𝐴) ∈ (Clsd‘𝐽))
2625adantr 482 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋𝐴) ∈ (Clsd‘𝐽))
2710, 26eqeltrrd 2835 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐵 ∈ (Clsd‘𝐽))
285, 16, 18, 20, 27connclo 22789 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐵 = 𝑋)
2928difeq2d 4086 . . . . . . 7 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋𝐵) = (𝑋𝑋))
30 difid 4334 . . . . . . 7 (𝑋𝑋) = ∅
3129, 30eqtrdi 2789 . . . . . 6 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋𝐵) = ∅)
3211, 14, 313eqtr3d 2781 . . . . 5 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐴 = ∅)
3332ex 414 . . . 4 (𝜑 → ((𝑋𝐴) = 𝐵𝐴 = ∅))
349, 33sylbid 239 . . 3 (𝜑 → ((𝐴𝐵) = 𝑋𝐴 = ∅))
3534necon3d 2961 . 2 (𝜑 → (𝐴 ≠ ∅ → (𝐴𝐵) ≠ 𝑋))
361, 35mpd 15 1 (𝜑 → (𝐴𝐵) ≠ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wne 2940  cdif 3911  cun 3912  cin 3913  wss 3914  c0 4286  {cpr 4592   cuni 4869  cfv 6500  Topctop 22265  Clsdccld 22390  Conncconn 22785
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-top 22266  df-cld 22393  df-conn 22786
This theorem is referenced by:  dfconn2  22793
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