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Theorem prnmaxl 6471
 Description: A lower cut has no largest member. (Contributed by Jim Kingdon, 29-Sep-2019.)
Assertion
Ref Expression
prnmaxl ((⟨𝐿, 𝑈 P B 𝐿) → x 𝐿 B <Q x)
Distinct variable groups:   x,B   x,𝐿   x,𝑈

Proof of Theorem prnmaxl
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 elprnql 6464 . . . . 5 ((⟨𝐿, 𝑈 P B 𝐿) → B Q)
2 elinp 6457 . . . . . . . 8 (⟨𝐿, 𝑈 P ↔ (((𝐿Q 𝑈Q) (y Q y 𝐿 x Q x 𝑈)) ((y Q (y 𝐿x Q (y <Q x x 𝐿)) x Q (x 𝑈y Q (y <Q x y 𝑈))) y Q ¬ (y 𝐿 y 𝑈) y Q x Q (y <Q x → (y 𝐿 x 𝑈)))))
3 simpr1l 960 . . . . . . . 8 ((((𝐿Q 𝑈Q) (y Q y 𝐿 x Q x 𝑈)) ((y Q (y 𝐿x Q (y <Q x x 𝐿)) x Q (x 𝑈y Q (y <Q x y 𝑈))) y Q ¬ (y 𝐿 y 𝑈) y Q x Q (y <Q x → (y 𝐿 x 𝑈)))) → y Q (y 𝐿x Q (y <Q x x 𝐿)))
42, 3sylbi 114 . . . . . . 7 (⟨𝐿, 𝑈 Py Q (y 𝐿x Q (y <Q x x 𝐿)))
5 eleq1 2097 . . . . . . . . 9 (y = B → (y 𝐿B 𝐿))
6 breq1 3758 . . . . . . . . . . 11 (y = B → (y <Q xB <Q x))
76anbi1d 438 . . . . . . . . . 10 (y = B → ((y <Q x x 𝐿) ↔ (B <Q x x 𝐿)))
87rexbidv 2321 . . . . . . . . 9 (y = B → (x Q (y <Q x x 𝐿) ↔ x Q (B <Q x x 𝐿)))
95, 8bibi12d 224 . . . . . . . 8 (y = B → ((y 𝐿x Q (y <Q x x 𝐿)) ↔ (B 𝐿x Q (B <Q x x 𝐿))))
109rspcv 2646 . . . . . . 7 (B Q → (y Q (y 𝐿x Q (y <Q x x 𝐿)) → (B 𝐿x Q (B <Q x x 𝐿))))
11 bi1 111 . . . . . . 7 ((B 𝐿x Q (B <Q x x 𝐿)) → (B 𝐿x Q (B <Q x x 𝐿)))
124, 10, 11syl56 30 . . . . . 6 (B Q → (⟨𝐿, 𝑈 P → (B 𝐿x Q (B <Q x x 𝐿))))
1312impd 242 . . . . 5 (B Q → ((⟨𝐿, 𝑈 P B 𝐿) → x Q (B <Q x x 𝐿)))
141, 13mpcom 32 . . . 4 ((⟨𝐿, 𝑈 P B 𝐿) → x Q (B <Q x x 𝐿))
15 df-rex 2306 . . . 4 (x Q (B <Q x x 𝐿) ↔ x(x Q (B <Q x x 𝐿)))
1614, 15sylib 127 . . 3 ((⟨𝐿, 𝑈 P B 𝐿) → x(x Q (B <Q x x 𝐿)))
17 ltrelnq 6349 . . . . . . . . 9 <Q ⊆ (Q × Q)
1817brel 4335 . . . . . . . 8 (B <Q x → (B Q x Q))
1918simprd 107 . . . . . . 7 (B <Q xx Q)
2019pm4.71ri 372 . . . . . 6 (B <Q x ↔ (x Q B <Q x))
2120anbi1i 431 . . . . 5 ((B <Q x x 𝐿) ↔ ((x Q B <Q x) x 𝐿))
22 ancom 253 . . . . 5 ((B <Q x x 𝐿) ↔ (x 𝐿 B <Q x))
23 anass 381 . . . . 5 (((x Q B <Q x) x 𝐿) ↔ (x Q (B <Q x x 𝐿)))
2421, 22, 233bitr3i 199 . . . 4 ((x 𝐿 B <Q x) ↔ (x Q (B <Q x x 𝐿)))
2524exbii 1493 . . 3 (x(x 𝐿 B <Q x) ↔ x(x Q (B <Q x x 𝐿)))
2616, 25sylibr 137 . 2 ((⟨𝐿, 𝑈 P B 𝐿) → x(x 𝐿 B <Q x))
27 df-rex 2306 . 2 (x 𝐿 B <Q xx(x 𝐿 B <Q x))
2826, 27sylibr 137 1 ((⟨𝐿, 𝑈 P B 𝐿) → x 𝐿 B <Q x)
 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
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