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Theorem prnminu 6644
 Description: An upper cut has no smallest member. (Contributed by Jim Kingdon, 7-Nov-2019.)
Assertion
Ref Expression
prnminu ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝑈) → ∃𝑥𝑈 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 6637 . . . . 5 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝑈) → 𝐵Q)
2 elinp 6629 . . . . . . . 8 (⟨𝐿, 𝑈⟩ ∈ P ↔ (((𝐿Q𝑈Q) ∧ (∃𝑥Q 𝑥𝐿 ∧ ∃𝑦Q 𝑦𝑈)) ∧ ((∀𝑥Q (𝑥𝐿 ↔ ∃𝑦Q (𝑥 <Q 𝑦𝑦𝐿)) ∧ ∀𝑦Q (𝑦𝑈 ↔ ∃𝑥Q (𝑥 <Q 𝑦𝑥𝑈))) ∧ ∀𝑥Q ¬ (𝑥𝐿𝑥𝑈) ∧ ∀𝑥Q𝑦Q (𝑥 <Q 𝑦 → (𝑥𝐿𝑦𝑈)))))
3 simpr1r 973 . . . . . . . 8 ((((𝐿Q𝑈Q) ∧ (∃𝑥Q 𝑥𝐿 ∧ ∃𝑦Q 𝑦𝑈)) ∧ ((∀𝑥Q (𝑥𝐿 ↔ ∃𝑦Q (𝑥 <Q 𝑦𝑦𝐿)) ∧ ∀𝑦Q (𝑦𝑈 ↔ ∃𝑥Q (𝑥 <Q 𝑦𝑥𝑈))) ∧ ∀𝑥Q ¬ (𝑥𝐿𝑥𝑈) ∧ ∀𝑥Q𝑦Q (𝑥 <Q 𝑦 → (𝑥𝐿𝑦𝑈)))) → ∀𝑦Q (𝑦𝑈 ↔ ∃𝑥Q (𝑥 <Q 𝑦𝑥𝑈)))
42, 3sylbi 118 . . . . . . 7 (⟨𝐿, 𝑈⟩ ∈ P → ∀𝑦Q (𝑦𝑈 ↔ ∃𝑥Q (𝑥 <Q 𝑦𝑥𝑈)))
5 eleq1 2116 . . . . . . . . 9 (𝑦 = 𝐵 → (𝑦𝑈𝐵𝑈))
6 breq2 3795 . . . . . . . . . . 11 (𝑦 = 𝐵 → (𝑥 <Q 𝑦𝑥 <Q 𝐵))
76anbi1d 446 . . . . . . . . . 10 (𝑦 = 𝐵 → ((𝑥 <Q 𝑦𝑥𝑈) ↔ (𝑥 <Q 𝐵𝑥𝑈)))
87rexbidv 2344 . . . . . . . . 9 (𝑦 = 𝐵 → (∃𝑥Q (𝑥 <Q 𝑦𝑥𝑈) ↔ ∃𝑥Q (𝑥 <Q 𝐵𝑥𝑈)))
95, 8bibi12d 228 . . . . . . . 8 (𝑦 = 𝐵 → ((𝑦𝑈 ↔ ∃𝑥Q (𝑥 <Q 𝑦𝑥𝑈)) ↔ (𝐵𝑈 ↔ ∃𝑥Q (𝑥 <Q 𝐵𝑥𝑈))))
109rspcv 2669 . . . . . . 7 (𝐵Q → (∀𝑦Q (𝑦𝑈 ↔ ∃𝑥Q (𝑥 <Q 𝑦𝑥𝑈)) → (𝐵𝑈 ↔ ∃𝑥Q (𝑥 <Q 𝐵𝑥𝑈))))
11 bi1 115 . . . . . . 7 ((𝐵𝑈 ↔ ∃𝑥Q (𝑥 <Q 𝐵𝑥𝑈)) → (𝐵𝑈 → ∃𝑥Q (𝑥 <Q 𝐵𝑥𝑈)))
124, 10, 11syl56 34 . . . . . 6 (𝐵Q → (⟨𝐿, 𝑈⟩ ∈ P → (𝐵𝑈 → ∃𝑥Q (𝑥 <Q 𝐵𝑥𝑈))))
1312impd 246 . . . . 5 (𝐵Q → ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝑈) → ∃𝑥Q (𝑥 <Q 𝐵𝑥𝑈)))
141, 13mpcom 36 . . . 4 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝑈) → ∃𝑥Q (𝑥 <Q 𝐵𝑥𝑈))
15 df-rex 2329 . . . 4 (∃𝑥Q (𝑥 <Q 𝐵𝑥𝑈) ↔ ∃𝑥(𝑥Q ∧ (𝑥 <Q 𝐵𝑥𝑈)))
1614, 15sylib 131 . . 3 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝑈) → ∃𝑥(𝑥Q ∧ (𝑥 <Q 𝐵𝑥𝑈)))
17 ltrelnq 6520 . . . . . . . . 9 <Q ⊆ (Q × Q)
1817brel 4419 . . . . . . . 8 (𝑥 <Q 𝐵 → (𝑥Q𝐵Q))
1918simpld 109 . . . . . . 7 (𝑥 <Q 𝐵𝑥Q)
2019pm4.71ri 378 . . . . . 6 (𝑥 <Q 𝐵 ↔ (𝑥Q𝑥 <Q 𝐵))
2120anbi1i 439 . . . . 5 ((𝑥 <Q 𝐵𝑥𝑈) ↔ ((𝑥Q𝑥 <Q 𝐵) ∧ 𝑥𝑈))
22 ancom 257 . . . . 5 ((𝑥 <Q 𝐵𝑥𝑈) ↔ (𝑥𝑈𝑥 <Q 𝐵))
23 anass 387 . . . . 5 (((𝑥Q𝑥 <Q 𝐵) ∧ 𝑥𝑈) ↔ (𝑥Q ∧ (𝑥 <Q 𝐵𝑥𝑈)))
2421, 22, 233bitr3i 203 . . . 4 ((𝑥𝑈𝑥 <Q 𝐵) ↔ (𝑥Q ∧ (𝑥 <Q 𝐵𝑥𝑈)))
2524exbii 1512 . . 3 (∃𝑥(𝑥𝑈𝑥 <Q 𝐵) ↔ ∃𝑥(𝑥Q ∧ (𝑥 <Q 𝐵𝑥𝑈)))
2616, 25sylibr 141 . 2 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝑈) → ∃𝑥(𝑥𝑈𝑥 <Q 𝐵))
27 df-rex 2329 . 2 (∃𝑥𝑈 𝑥 <Q 𝐵 ↔ ∃𝑥(𝑥𝑈𝑥 <Q 𝐵))
2826, 27sylibr 141 1 ((⟨𝐿, 𝑈⟩ ∈ P𝐵𝑈) → ∃𝑥𝑈 𝑥 <Q 𝐵)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 101   ↔ wb 102   ∨ wo 639   ∧ w3a 896   = wceq 1259  ∃wex 1397   ∈ wcel 1409  ∀wral 2323  ∃wrex 2324   ⊆ wss 2944  ⟨cop 3405   class class class wbr 3791  Qcnq 6435
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