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Theorem prmu 7308
 Description: A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
Assertion
Ref Expression
prmu (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥Q 𝑥𝑈)
Distinct variable groups:   𝑥,𝐿   𝑥,𝑈

Proof of Theorem prmu
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elinp 7304 . 2 (⟨𝐿, 𝑈⟩ ∈ P ↔ (((𝐿Q𝑈Q) ∧ (∃𝑦Q 𝑦𝐿 ∧ ∃𝑥Q 𝑥𝑈)) ∧ ((∀𝑦Q (𝑦𝐿 ↔ ∃𝑥Q (𝑦 <Q 𝑥𝑥𝐿)) ∧ ∀𝑥Q (𝑥𝑈 ↔ ∃𝑦Q (𝑦 <Q 𝑥𝑦𝑈))) ∧ ∀𝑦Q ¬ (𝑦𝐿𝑦𝑈) ∧ ∀𝑦Q𝑥Q (𝑦 <Q 𝑥 → (𝑦𝐿𝑥𝑈)))))
2 simplrr 526 . 2 ((((𝐿Q𝑈Q) ∧ (∃𝑦Q 𝑦𝐿 ∧ ∃𝑥Q 𝑥𝑈)) ∧ ((∀𝑦Q (𝑦𝐿 ↔ ∃𝑥Q (𝑦 <Q 𝑥𝑥𝐿)) ∧ ∀𝑥Q (𝑥𝑈 ↔ ∃𝑦Q (𝑦 <Q 𝑥𝑦𝑈))) ∧ ∀𝑦Q ¬ (𝑦𝐿𝑦𝑈) ∧ ∀𝑦Q𝑥Q (𝑦 <Q 𝑥 → (𝑦𝐿𝑥𝑈)))) → ∃𝑥Q 𝑥𝑈)
31, 2sylbi 120 1 (⟨𝐿, 𝑈⟩ ∈ P → ∃𝑥Q 𝑥𝑈)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ∧ w3a 963   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418   ⊆ wss 3074  ⟨cop 3533   class class class wbr 3935  Qcnq 7110
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