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Theorem conndisj 21200
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 4458 . . . . . . 7 (𝐴𝐽𝐴 𝐽)
42, 3syl 17 . . . . . 6 (𝜑𝐴 𝐽)
5 isconn.1 . . . . . 6 𝑋 = 𝐽
64, 5syl6sseqr 3644 . . . . 5 (𝜑𝐴𝑋)
7 conndisj.6 . . . . 5 (𝜑 → (𝐴𝐵) = ∅)
8 uneqdifeq 4048 . . . . 5 ((𝐴𝑋 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) = 𝑋 ↔ (𝑋𝐴) = 𝐵))
96, 7, 8syl2anc 692 . . . 4 (𝜑 → ((𝐴𝐵) = 𝑋 ↔ (𝑋𝐴) = 𝐵))
10 simpr 477 . . . . . . 7 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋𝐴) = 𝐵)
1110difeq2d 3720 . . . . . 6 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋 ∖ (𝑋𝐴)) = (𝑋𝐵))
12 dfss4 3850 . . . . . . . 8 (𝐴𝑋 ↔ (𝑋 ∖ (𝑋𝐴)) = 𝐴)
136, 12sylib 208 . . . . . . 7 (𝜑 → (𝑋 ∖ (𝑋𝐴)) = 𝐴)
1413adantr 481 . . . . . 6 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋 ∖ (𝑋𝐴)) = 𝐴)
15 connclo.1 . . . . . . . . . 10 (𝜑𝐽 ∈ Conn)
1615adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐽 ∈ Conn)
17 conndisj.4 . . . . . . . . . 10 (𝜑𝐵𝐽)
1817adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐵𝐽)
19 conndisj.5 . . . . . . . . . 10 (𝜑𝐵 ≠ ∅)
2019adantr 481 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐵 ≠ ∅)
215isconn 21197 . . . . . . . . . . . . . 14 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
2221simplbi 476 . . . . . . . . . . . . 13 (𝐽 ∈ Conn → 𝐽 ∈ Top)
2315, 22syl 17 . . . . . . . . . . . 12 (𝜑𝐽 ∈ Top)
245opncld 20818 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐴𝐽) → (𝑋𝐴) ∈ (Clsd‘𝐽))
2523, 2, 24syl2anc 692 . . . . . . . . . . 11 (𝜑 → (𝑋𝐴) ∈ (Clsd‘𝐽))
2625adantr 481 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋𝐴) ∈ (Clsd‘𝐽))
2710, 26eqeltrrd 2700 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐵 ∈ (Clsd‘𝐽))
285, 16, 18, 20, 27connclo 21199 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐵 = 𝑋)
2928difeq2d 3720 . . . . . . 7 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋𝐵) = (𝑋𝑋))
30 difid 3939 . . . . . . 7 (𝑋𝑋) = ∅
3129, 30syl6eq 2670 . . . . . 6 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋𝐵) = ∅)
3211, 14, 313eqtr3d 2662 . . . . 5 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐴 = ∅)
3332ex 450 . . . 4 (𝜑 → ((𝑋𝐴) = 𝐵𝐴 = ∅))
349, 33sylbid 230 . . 3 (𝜑 → ((𝐴𝐵) = 𝑋𝐴 = ∅))
3534necon3d 2812 . 2 (𝜑 → (𝐴 ≠ ∅ → (𝐴𝐵) ≠ 𝑋))
361, 35mpd 15 1 (𝜑 → (𝐴𝐵) ≠ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wne 2791  cdif 3564  cun 3565  cin 3566  wss 3567  c0 3907  {cpr 4170   cuni 4427  cfv 5876  Topctop 20679  Clsdccld 20801  Conncconn 21195
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-top 20680  df-cld 20804  df-conn 21196
This theorem is referenced by:  dfconn2  21203
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