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 Description: A positive real's upper cut is a subset of the positive fractions. It would presumably be possible to also prove 𝑈 ⊊ Q, but we only need 𝑈 ⊆ Q so far. (Contributed by Jim Kingdon, 28-Sep-2019.)
Assertion
Ref Expression

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elinp 6553 . 2 (⟨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)))))
2 simpllr 486 . 2 ((((𝐿Q𝑈Q) ∧ (∃𝑥Q 𝑥𝐿 ∧ ∃𝑦Q 𝑦𝑈)) ∧ ((∀𝑥Q (𝑥𝐿 ↔ ∃𝑦Q (𝑥 <Q 𝑦𝑦𝐿)) ∧ ∀𝑦Q (𝑦𝑈 ↔ ∃𝑥Q (𝑥 <Q 𝑦𝑥𝑈))) ∧ ∀𝑥Q ¬ (𝑥𝐿𝑥𝑈) ∧ ∀𝑥Q𝑦Q (𝑥 <Q 𝑦 → (𝑥𝐿𝑦𝑈)))) → 𝑈Q)
31, 2sylbi 114 1 (⟨𝐿, 𝑈⟩ ∈ P𝑈Q)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   ∧ w3a 885   ∈ wcel 1393  ∀wral 2303  ∃wrex 2304   ⊆ wss 2914  ⟨cop 3375   class class class wbr 3761  Qcnq 6359
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