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Theorem prltlu 6585
 Description: An element of a lower cut is less than an element of the corresponding upper cut. (Contributed by Jim Kingdon, 15-Oct-2019.)
Assertion
Ref Expression
prltlu ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → 𝐵 <Q 𝐶)

Proof of Theorem prltlu
Dummy variables 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 906 . . 3 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → 𝐶𝑈)
2 elprnqu 6580 . . . . . 6 ((⟨𝐿, 𝑈⟩ ∈ P𝐶𝑈) → 𝐶Q)
323adant2 923 . . . . 5 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → 𝐶Q)
4 elinp 6572 . . . . . . 7 (⟨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)))))
5 simpr2 911 . . . . . . 7 ((((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))
64, 5sylbi 114 . . . . . 6 (⟨𝐿, 𝑈⟩ ∈ P → ∀𝑞Q ¬ (𝑞𝐿𝑞𝑈))
763ad2ant1 925 . . . . 5 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → ∀𝑞Q ¬ (𝑞𝐿𝑞𝑈))
8 eleq1 2100 . . . . . . . 8 (𝑞 = 𝐶 → (𝑞𝐿𝐶𝐿))
9 eleq1 2100 . . . . . . . 8 (𝑞 = 𝐶 → (𝑞𝑈𝐶𝑈))
108, 9anbi12d 442 . . . . . . 7 (𝑞 = 𝐶 → ((𝑞𝐿𝑞𝑈) ↔ (𝐶𝐿𝐶𝑈)))
1110notbid 592 . . . . . 6 (𝑞 = 𝐶 → (¬ (𝑞𝐿𝑞𝑈) ↔ ¬ (𝐶𝐿𝐶𝑈)))
1211rspcv 2652 . . . . 5 (𝐶Q → (∀𝑞Q ¬ (𝑞𝐿𝑞𝑈) → ¬ (𝐶𝐿𝐶𝑈)))
133, 7, 12sylc 56 . . . 4 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → ¬ (𝐶𝐿𝐶𝑈))
14 ancom 253 . . . . . 6 ((𝐶𝐿𝐶𝑈) ↔ (𝐶𝑈𝐶𝐿))
1514notbii 594 . . . . 5 (¬ (𝐶𝐿𝐶𝑈) ↔ ¬ (𝐶𝑈𝐶𝐿))
16 imnan 624 . . . . 5 ((𝐶𝑈 → ¬ 𝐶𝐿) ↔ ¬ (𝐶𝑈𝐶𝐿))
1715, 16bitr4i 176 . . . 4 (¬ (𝐶𝐿𝐶𝑈) ↔ (𝐶𝑈 → ¬ 𝐶𝐿))
1813, 17sylib 127 . . 3 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → (𝐶𝑈 → ¬ 𝐶𝐿))
191, 18mpd 13 . 2 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → ¬ 𝐶𝐿)
20 3simpa 901 . . 3 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → (⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿))
21 prubl 6584 . . 3 (((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿) ∧ 𝐶Q) → (¬ 𝐶𝐿𝐵 <Q 𝐶))
2220, 3, 21syl2anc 391 . 2 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → (¬ 𝐶𝐿𝐵 <Q 𝐶))
2319, 22mpd 13 1 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → 𝐵 <Q 𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307   ⊆ wss 2917  ⟨cop 3378   class class class wbr 3764  Qcnq 6378
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