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Theorem conndisj 22024
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 4833 . . . . . . 7 (𝐴𝐽𝐴 𝐽)
42, 3syl 17 . . . . . 6 (𝜑𝐴 𝐽)
5 isconn.1 . . . . . 6 𝑋 = 𝐽
64, 5sseqtrrdi 3969 . . . . 5 (𝜑𝐴𝑋)
7 conndisj.6 . . . . 5 (𝜑 → (𝐴𝐵) = ∅)
8 uneqdifeq 4399 . . . . 5 ((𝐴𝑋 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) = 𝑋 ↔ (𝑋𝐴) = 𝐵))
96, 7, 8syl2anc 587 . . . 4 (𝜑 → ((𝐴𝐵) = 𝑋 ↔ (𝑋𝐴) = 𝐵))
10 simpr 488 . . . . . . 7 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋𝐴) = 𝐵)
1110difeq2d 4053 . . . . . 6 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋 ∖ (𝑋𝐴)) = (𝑋𝐵))
12 dfss4 4188 . . . . . . . 8 (𝐴𝑋 ↔ (𝑋 ∖ (𝑋𝐴)) = 𝐴)
136, 12sylib 221 . . . . . . 7 (𝜑 → (𝑋 ∖ (𝑋𝐴)) = 𝐴)
1413adantr 484 . . . . . 6 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋 ∖ (𝑋𝐴)) = 𝐴)
15 connclo.1 . . . . . . . . . 10 (𝜑𝐽 ∈ Conn)
1615adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐽 ∈ Conn)
17 conndisj.4 . . . . . . . . . 10 (𝜑𝐵𝐽)
1817adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐵𝐽)
19 conndisj.5 . . . . . . . . . 10 (𝜑𝐵 ≠ ∅)
2019adantr 484 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐵 ≠ ∅)
215isconn 22021 . . . . . . . . . . . . . 14 (𝐽 ∈ Conn ↔ (𝐽 ∈ Top ∧ (𝐽 ∩ (Clsd‘𝐽)) = {∅, 𝑋}))
2221simplbi 501 . . . . . . . . . . . . 13 (𝐽 ∈ Conn → 𝐽 ∈ Top)
2315, 22syl 17 . . . . . . . . . . . 12 (𝜑𝐽 ∈ Top)
245opncld 21641 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝐴𝐽) → (𝑋𝐴) ∈ (Clsd‘𝐽))
2523, 2, 24syl2anc 587 . . . . . . . . . . 11 (𝜑 → (𝑋𝐴) ∈ (Clsd‘𝐽))
2625adantr 484 . . . . . . . . . 10 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋𝐴) ∈ (Clsd‘𝐽))
2710, 26eqeltrrd 2894 . . . . . . . . 9 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐵 ∈ (Clsd‘𝐽))
285, 16, 18, 20, 27connclo 22023 . . . . . . . 8 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐵 = 𝑋)
2928difeq2d 4053 . . . . . . 7 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋𝐵) = (𝑋𝑋))
30 difid 4287 . . . . . . 7 (𝑋𝑋) = ∅
3129, 30eqtrdi 2852 . . . . . 6 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → (𝑋𝐵) = ∅)
3211, 14, 313eqtr3d 2844 . . . . 5 ((𝜑 ∧ (𝑋𝐴) = 𝐵) → 𝐴 = ∅)
3332ex 416 . . . 4 (𝜑 → ((𝑋𝐴) = 𝐵𝐴 = ∅))
349, 33sylbid 243 . . 3 (𝜑 → ((𝐴𝐵) = 𝑋𝐴 = ∅))
3534necon3d 3011 . 2 (𝜑 → (𝐴 ≠ ∅ → (𝐴𝐵) ≠ 𝑋))
361, 35mpd 15 1 (𝜑 → (𝐴𝐵) ≠ 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  wne 2990  cdif 3881  cun 3882  cin 3883  wss 3884  c0 4246  {cpr 4530   cuni 4803  cfv 6328  Topctop 21501  Clsdccld 21624  Conncconn 22019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-iota 6287  df-fun 6330  df-fv 6336  df-top 21502  df-cld 21627  df-conn 22020
This theorem is referenced by:  dfconn2  22027
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