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Theorem prdisj 7323
 Description: A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.)
Assertion
Ref Expression
prdisj ((⟨𝐿, 𝑈⟩ ∈ P𝐴Q) → ¬ (𝐴𝐿𝐴𝑈))

Proof of Theorem prdisj
Dummy variables 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2203 . . . . 5 (𝑞 = 𝐴 → (𝑞Q𝐴Q))
21anbi2d 460 . . . 4 (𝑞 = 𝐴 → ((⟨𝐿, 𝑈⟩ ∈ P𝑞Q) ↔ (⟨𝐿, 𝑈⟩ ∈ P𝐴Q)))
3 eleq1 2203 . . . . . 6 (𝑞 = 𝐴 → (𝑞𝐿𝐴𝐿))
4 eleq1 2203 . . . . . 6 (𝑞 = 𝐴 → (𝑞𝑈𝐴𝑈))
53, 4anbi12d 465 . . . . 5 (𝑞 = 𝐴 → ((𝑞𝐿𝑞𝑈) ↔ (𝐴𝐿𝐴𝑈)))
65notbid 657 . . . 4 (𝑞 = 𝐴 → (¬ (𝑞𝐿𝑞𝑈) ↔ ¬ (𝐴𝐿𝐴𝑈)))
72, 6imbi12d 233 . . 3 (𝑞 = 𝐴 → (((⟨𝐿, 𝑈⟩ ∈ P𝑞Q) → ¬ (𝑞𝐿𝑞𝑈)) ↔ ((⟨𝐿, 𝑈⟩ ∈ P𝐴Q) → ¬ (𝐴𝐿𝐴𝑈))))
8 elinp 7305 . . . . 5 (⟨𝐿, 𝑈⟩ ∈ P ↔ (((𝐿Q𝑈Q) ∧ (∃𝑞Q 𝑞𝐿 ∧ ∃𝑟Q 𝑟𝑈)) ∧ ((∀𝑞Q (𝑞𝐿 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟𝐿)) ∧ ∀𝑟Q (𝑟𝑈 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞𝑈))) ∧ ∀𝑞Q ¬ (𝑞𝐿𝑞𝑈) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞𝐿𝑟𝑈)))))
9 simpr2 989 . . . . 5 ((((𝐿Q𝑈Q) ∧ (∃𝑞Q 𝑞𝐿 ∧ ∃𝑟Q 𝑟𝑈)) ∧ ((∀𝑞Q (𝑞𝐿 ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟𝐿)) ∧ ∀𝑟Q (𝑟𝑈 ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞𝑈))) ∧ ∀𝑞Q ¬ (𝑞𝐿𝑞𝑈) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞𝐿𝑟𝑈)))) → ∀𝑞Q ¬ (𝑞𝐿𝑞𝑈))
108, 9sylbi 120 . . . 4 (⟨𝐿, 𝑈⟩ ∈ P → ∀𝑞Q ¬ (𝑞𝐿𝑞𝑈))
1110r19.21bi 2523 . . 3 ((⟨𝐿, 𝑈⟩ ∈ P𝑞Q) → ¬ (𝑞𝐿𝑞𝑈))
127, 11vtoclg 2749 . 2 (𝐴Q → ((⟨𝐿, 𝑈⟩ ∈ P𝐴Q) → ¬ (𝐴𝐿𝐴𝑈)))
1312anabsi7 571 1 ((⟨𝐿, 𝑈⟩ ∈ P𝐴Q) → ¬ (𝐴𝐿𝐴𝑈))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ∧ w3a 963   = wceq 1332   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418   ⊆ wss 3075  ⟨cop 3534   class class class wbr 3936  Qcnq 7111
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