Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  prltlu Structured version   GIF version

Theorem prltlu 6341
 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 P B 𝐿 𝐶 𝑈) → B <Q 𝐶)

Proof of Theorem prltlu
Dummy variables 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp3 894 . . 3 ((⟨𝐿, 𝑈 P B 𝐿 𝐶 𝑈) → 𝐶 𝑈)
2 elprnqu 6336 . . . . . 6 ((⟨𝐿, 𝑈 P 𝐶 𝑈) → 𝐶 Q)
323adant2 911 . . . . 5 ((⟨𝐿, 𝑈 P B 𝐿 𝐶 𝑈) → 𝐶 Q)
4 elinp 6328 . . . . . . 7 (⟨𝐿, 𝑈 P ↔ (((𝐿Q 𝑈Q) (𝑞 Q 𝑞 𝐿 𝑟 Q 𝑟 𝑈)) ((𝑞 Q (𝑞 𝐿𝑟 Q (𝑞 <Q 𝑟 𝑟 𝐿)) 𝑟 Q (𝑟 𝑈𝑞 Q (𝑞 <Q 𝑟 𝑞 𝑈))) 𝑞 Q ¬ (𝑞 𝐿 𝑞 𝑈) 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 𝐿 𝑟 𝑈)))))
5 simpr2 899 . . . . . . 7 ((((𝐿Q 𝑈Q) (𝑞 Q 𝑞 𝐿 𝑟 Q 𝑟 𝑈)) ((𝑞 Q (𝑞 𝐿𝑟 Q (𝑞 <Q 𝑟 𝑟 𝐿)) 𝑟 Q (𝑟 𝑈𝑞 Q (𝑞 <Q 𝑟 𝑞 𝑈))) 𝑞 Q ¬ (𝑞 𝐿 𝑞 𝑈) 𝑞 Q 𝑟 Q (𝑞 <Q 𝑟 → (𝑞 𝐿 𝑟 𝑈)))) → 𝑞 Q ¬ (𝑞 𝐿 𝑞 𝑈))
64, 5sylbi 114 . . . . . 6 (⟨𝐿, 𝑈 P𝑞 Q ¬ (𝑞 𝐿 𝑞 𝑈))
763ad2ant1 913 . . . . 5 ((⟨𝐿, 𝑈 P B 𝐿 𝐶 𝑈) → 𝑞 Q ¬ (𝑞 𝐿 𝑞 𝑈))
8 eleq1 2082 . . . . . . . 8 (𝑞 = 𝐶 → (𝑞 𝐿𝐶 𝐿))
9 eleq1 2082 . . . . . . . 8 (𝑞 = 𝐶 → (𝑞 𝑈𝐶 𝑈))
108, 9anbi12d 445 . . . . . . 7 (𝑞 = 𝐶 → ((𝑞 𝐿 𝑞 𝑈) ↔ (𝐶 𝐿 𝐶 𝑈)))
1110notbid 579 . . . . . 6 (𝑞 = 𝐶 → (¬ (𝑞 𝐿 𝑞 𝑈) ↔ ¬ (𝐶 𝐿 𝐶 𝑈)))
1211rspcv 2629 . . . . 5 (𝐶 Q → (𝑞 Q ¬ (𝑞 𝐿 𝑞 𝑈) → ¬ (𝐶 𝐿 𝐶 𝑈)))
133, 7, 12sylc 56 . . . 4 ((⟨𝐿, 𝑈 P B 𝐿 𝐶 𝑈) → ¬ (𝐶 𝐿 𝐶 𝑈))
14 ancom 253 . . . . . 6 ((𝐶 𝐿 𝐶 𝑈) ↔ (𝐶 𝑈 𝐶 𝐿))
1514notbii 581 . . . . 5 (¬ (𝐶 𝐿 𝐶 𝑈) ↔ ¬ (𝐶 𝑈 𝐶 𝐿))
16 imnan 611 . . . . 5 ((𝐶 𝑈 → ¬ 𝐶 𝐿) ↔ ¬ (𝐶 𝑈 𝐶 𝐿))
1715, 16bitr4i 176 . . . 4 (¬ (𝐶 𝐿 𝐶 𝑈) ↔ (𝐶 𝑈 → ¬ 𝐶 𝐿))
1813, 17sylib 127 . . 3 ((⟨𝐿, 𝑈 P B 𝐿 𝐶 𝑈) → (𝐶 𝑈 → ¬ 𝐶 𝐿))
191, 18mpd 13 . 2 ((⟨𝐿, 𝑈 P B 𝐿 𝐶 𝑈) → ¬ 𝐶 𝐿)
20 3simpa 889 . . 3 ((⟨𝐿, 𝑈 P B 𝐿 𝐶 𝑈) → (⟨𝐿, 𝑈 P B 𝐿))
21 prubl 6340 . . 3 (((⟨𝐿, 𝑈 P B 𝐿) 𝐶 Q) → (¬ 𝐶 𝐿B <Q 𝐶))
2220, 3, 21syl2anc 393 . 2 ((⟨𝐿, 𝑈 P B 𝐿 𝐶 𝑈) → (¬ 𝐶 𝐿B <Q 𝐶))
2319, 22mpd 13 1 ((⟨𝐿, 𝑈 P B 𝐿 𝐶 𝑈) → B <Q 𝐶)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 616   ∧ w3a 873   = wceq 1228   ∈ wcel 1374  ∀wral 2284  ∃wrex 2285   ⊆ wss 2894  ⟨cop 3353   class class class wbr 3738  Qcnq 6138
 Copyright terms: Public domain W3C validator