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Theorem prltlu 7196
 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 951 . . 3 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → 𝐶𝑈)
2 eleq1 2162 . . . . . . 7 (𝑞 = 𝐶 → (𝑞𝐿𝐶𝐿))
3 eleq1 2162 . . . . . . 7 (𝑞 = 𝐶 → (𝑞𝑈𝐶𝑈))
42, 3anbi12d 460 . . . . . 6 (𝑞 = 𝐶 → ((𝑞𝐿𝑞𝑈) ↔ (𝐶𝐿𝐶𝑈)))
54notbid 633 . . . . 5 (𝑞 = 𝐶 → (¬ (𝑞𝐿𝑞𝑈) ↔ ¬ (𝐶𝐿𝐶𝑈)))
6 elinp 7183 . . . . . . 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)))))
7 simpr2 956 . . . . . . 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))
86, 7sylbi 120 . . . . . 6 (⟨𝐿, 𝑈⟩ ∈ P → ∀𝑞Q ¬ (𝑞𝐿𝑞𝑈))
983ad2ant1 970 . . . . 5 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → ∀𝑞Q ¬ (𝑞𝐿𝑞𝑈))
10 elprnqu 7191 . . . . . 6 ((⟨𝐿, 𝑈⟩ ∈ P𝐶𝑈) → 𝐶Q)
11103adant2 968 . . . . 5 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → 𝐶Q)
125, 9, 11rspcdva 2749 . . . 4 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → ¬ (𝐶𝐿𝐶𝑈))
13 ancom 264 . . . . . 6 ((𝐶𝐿𝐶𝑈) ↔ (𝐶𝑈𝐶𝐿))
1413notbii 635 . . . . 5 (¬ (𝐶𝐿𝐶𝑈) ↔ ¬ (𝐶𝑈𝐶𝐿))
15 imnan 665 . . . . 5 ((𝐶𝑈 → ¬ 𝐶𝐿) ↔ ¬ (𝐶𝑈𝐶𝐿))
1614, 15bitr4i 186 . . . 4 (¬ (𝐶𝐿𝐶𝑈) ↔ (𝐶𝑈 → ¬ 𝐶𝐿))
1712, 16sylib 121 . . 3 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → (𝐶𝑈 → ¬ 𝐶𝐿))
181, 17mpd 13 . 2 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → ¬ 𝐶𝐿)
19 3simpa 946 . . 3 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → (⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿))
20 prubl 7195 . . 3 (((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿) ∧ 𝐶Q) → (¬ 𝐶𝐿𝐵 <Q 𝐶))
2119, 11, 20syl2anc 406 . 2 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → (¬ 𝐶𝐿𝐵 <Q 𝐶))
2218, 21mpd 13 1 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝐿𝐶𝑈) → 𝐵 <Q 𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 670   ∧ w3a 930   = wceq 1299   ∈ wcel 1448  ∀wral 2375  ∃wrex 2376   ⊆ wss 3021  ⟨cop 3477   class class class wbr 3875  Qcnq 6989
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