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

Theorem prnminu 6471
 Description: An upper cut has no smallest member. (Contributed by Jim Kingdon, 7-Nov-2019.)
Assertion
Ref Expression
prnminu ((⟨𝐿, 𝑈 P B 𝑈) → x 𝑈 x <Q B)
Distinct variable groups:   x,B   x,𝐿   x,𝑈

Proof of Theorem prnminu
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 elprnqu 6464 . . . . 5 ((⟨𝐿, 𝑈 P B 𝑈) → B Q)
2 elinp 6456 . . . . . . . 8 (⟨𝐿, 𝑈 P ↔ (((𝐿Q 𝑈Q) (x Q x 𝐿 y Q y 𝑈)) ((x Q (x 𝐿y Q (x <Q y y 𝐿)) y Q (y 𝑈x Q (x <Q y x 𝑈))) x Q ¬ (x 𝐿 x 𝑈) x Q y Q (x <Q y → (x 𝐿 y 𝑈)))))
3 simpr1r 961 . . . . . . . 8 ((((𝐿Q 𝑈Q) (x Q x 𝐿 y Q y 𝑈)) ((x Q (x 𝐿y Q (x <Q y y 𝐿)) y Q (y 𝑈x Q (x <Q y x 𝑈))) x Q ¬ (x 𝐿 x 𝑈) x Q y Q (x <Q y → (x 𝐿 y 𝑈)))) → y Q (y 𝑈x Q (x <Q y x 𝑈)))
42, 3sylbi 114 . . . . . . 7 (⟨𝐿, 𝑈 Py Q (y 𝑈x Q (x <Q y x 𝑈)))
5 eleq1 2097 . . . . . . . . 9 (y = B → (y 𝑈B 𝑈))
6 breq2 3759 . . . . . . . . . . 11 (y = B → (x <Q yx <Q B))
76anbi1d 438 . . . . . . . . . 10 (y = B → ((x <Q y x 𝑈) ↔ (x <Q B x 𝑈)))
87rexbidv 2321 . . . . . . . . 9 (y = B → (x Q (x <Q y x 𝑈) ↔ x Q (x <Q B x 𝑈)))
95, 8bibi12d 224 . . . . . . . 8 (y = B → ((y 𝑈x Q (x <Q y x 𝑈)) ↔ (B 𝑈x Q (x <Q B x 𝑈))))
109rspcv 2646 . . . . . . 7 (B Q → (y Q (y 𝑈x Q (x <Q y x 𝑈)) → (B 𝑈x Q (x <Q B x 𝑈))))
11 bi1 111 . . . . . . 7 ((B 𝑈x Q (x <Q B x 𝑈)) → (B 𝑈x Q (x <Q B x 𝑈)))
124, 10, 11syl56 30 . . . . . 6 (B Q → (⟨𝐿, 𝑈 P → (B 𝑈x Q (x <Q B x 𝑈))))
1312impd 242 . . . . 5 (B Q → ((⟨𝐿, 𝑈 P B 𝑈) → x Q (x <Q B x 𝑈)))
141, 13mpcom 32 . . . 4 ((⟨𝐿, 𝑈 P B 𝑈) → x Q (x <Q B x 𝑈))
15 df-rex 2306 . . . 4 (x Q (x <Q B x 𝑈) ↔ x(x Q (x <Q B x 𝑈)))
1614, 15sylib 127 . . 3 ((⟨𝐿, 𝑈 P B 𝑈) → x(x Q (x <Q B x 𝑈)))
17 ltrelnq 6349 . . . . . . . . 9 <Q ⊆ (Q × Q)
1817brel 4335 . . . . . . . 8 (x <Q B → (x Q B Q))
1918simpld 105 . . . . . . 7 (x <Q Bx Q)
2019pm4.71ri 372 . . . . . 6 (x <Q B ↔ (x Q x <Q B))
2120anbi1i 431 . . . . 5 ((x <Q B x 𝑈) ↔ ((x Q x <Q B) x 𝑈))
22 ancom 253 . . . . 5 ((x <Q B x 𝑈) ↔ (x 𝑈 x <Q B))
23 anass 381 . . . . 5 (((x Q x <Q B) x 𝑈) ↔ (x Q (x <Q B x 𝑈)))
2421, 22, 233bitr3i 199 . . . 4 ((x 𝑈 x <Q B) ↔ (x Q (x <Q B x 𝑈)))
2524exbii 1493 . . 3 (x(x 𝑈 x <Q B) ↔ x(x Q (x <Q B x 𝑈)))
2616, 25sylibr 137 . 2 ((⟨𝐿, 𝑈 P B 𝑈) → x(x 𝑈 x <Q B))
27 df-rex 2306 . 2 (x 𝑈 x <Q Bx(x 𝑈 x <Q B))
2826, 27sylibr 137 1 ((⟨𝐿, 𝑈 P B 𝑈) → x 𝑈 x <Q B)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628   ∧ w3a 884   = wceq 1242  ∃wex 1378   ∈ wcel 1390  ∀wral 2300  ∃wrex 2301   ⊆ wss 2911  ⟨cop 3370   class class class wbr 3755  Qcnq 6264
 Copyright terms: Public domain W3C validator