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

Theorem nqprxx 7366
 Description: The canonical embedding of the rationals into the reals, expressed with the same variable for the lower and upper cuts. (Contributed by Jim Kingdon, 8-Dec-2019.)
Assertion
Ref Expression
nqprxx (𝐴Q → ⟨{𝑥𝑥 <Q 𝐴}, {𝑥𝐴 <Q 𝑥}⟩ ∈ P)
Distinct variable group:   𝑥,𝐴

Proof of Theorem nqprxx
Dummy variables 𝑟 𝑞 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nqprm 7362 . . 3 (𝐴Q → (∃𝑞Q 𝑞 ∈ {𝑥𝑥 <Q 𝐴} ∧ ∃𝑟Q 𝑟 ∈ {𝑥𝐴 <Q 𝑥}))
2 ltrelnq 7185 . . . . . . 7 <Q ⊆ (Q × Q)
32brel 4591 . . . . . 6 (𝑥 <Q 𝐴 → (𝑥Q𝐴Q))
43simpld 111 . . . . 5 (𝑥 <Q 𝐴𝑥Q)
54abssi 3172 . . . 4 {𝑥𝑥 <Q 𝐴} ⊆ Q
62brel 4591 . . . . . 6 (𝐴 <Q 𝑥 → (𝐴Q𝑥Q))
76simprd 113 . . . . 5 (𝐴 <Q 𝑥𝑥Q)
87abssi 3172 . . . 4 {𝑥𝐴 <Q 𝑥} ⊆ Q
95, 8pm3.2i 270 . . 3 ({𝑥𝑥 <Q 𝐴} ⊆ Q ∧ {𝑥𝐴 <Q 𝑥} ⊆ Q)
101, 9jctil 310 . 2 (𝐴Q → (({𝑥𝑥 <Q 𝐴} ⊆ Q ∧ {𝑥𝐴 <Q 𝑥} ⊆ Q) ∧ (∃𝑞Q 𝑞 ∈ {𝑥𝑥 <Q 𝐴} ∧ ∃𝑟Q 𝑟 ∈ {𝑥𝐴 <Q 𝑥})))
11 nqprrnd 7363 . . 3 (𝐴Q → (∀𝑞Q (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ {𝑥𝑥 <Q 𝐴})) ∧ ∀𝑟Q (𝑟 ∈ {𝑥𝐴 <Q 𝑥} ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ {𝑥𝐴 <Q 𝑥}))))
12 nqprdisj 7364 . . 3 (𝐴Q → ∀𝑞Q ¬ (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ∧ 𝑞 ∈ {𝑥𝐴 <Q 𝑥}))
13 nqprloc 7365 . . 3 (𝐴Q → ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ∨ 𝑟 ∈ {𝑥𝐴 <Q 𝑥})))
1411, 12, 133jca 1161 . 2 (𝐴Q → ((∀𝑞Q (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ {𝑥𝑥 <Q 𝐴})) ∧ ∀𝑟Q (𝑟 ∈ {𝑥𝐴 <Q 𝑥} ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ {𝑥𝐴 <Q 𝑥}))) ∧ ∀𝑞Q ¬ (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ∧ 𝑞 ∈ {𝑥𝐴 <Q 𝑥}) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ∨ 𝑟 ∈ {𝑥𝐴 <Q 𝑥}))))
15 elinp 7294 . 2 (⟨{𝑥𝑥 <Q 𝐴}, {𝑥𝐴 <Q 𝑥}⟩ ∈ P ↔ ((({𝑥𝑥 <Q 𝐴} ⊆ Q ∧ {𝑥𝐴 <Q 𝑥} ⊆ Q) ∧ (∃𝑞Q 𝑞 ∈ {𝑥𝑥 <Q 𝐴} ∧ ∃𝑟Q 𝑟 ∈ {𝑥𝐴 <Q 𝑥})) ∧ ((∀𝑞Q (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ↔ ∃𝑟Q (𝑞 <Q 𝑟𝑟 ∈ {𝑥𝑥 <Q 𝐴})) ∧ ∀𝑟Q (𝑟 ∈ {𝑥𝐴 <Q 𝑥} ↔ ∃𝑞Q (𝑞 <Q 𝑟𝑞 ∈ {𝑥𝐴 <Q 𝑥}))) ∧ ∀𝑞Q ¬ (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ∧ 𝑞 ∈ {𝑥𝐴 <Q 𝑥}) ∧ ∀𝑞Q𝑟Q (𝑞 <Q 𝑟 → (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ∨ 𝑟 ∈ {𝑥𝐴 <Q 𝑥})))))
1610, 14, 15sylanbrc 413 1 (𝐴Q → ⟨{𝑥𝑥 <Q 𝐴}, {𝑥𝐴 <Q 𝑥}⟩ ∈ P)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697   ∧ w3a 962   ∈ wcel 1480  {cab 2125  ∀wral 2416  ∃wrex 2417   ⊆ wss 3071  ⟨cop 3530   class class class wbr 3929  Qcnq 7100
 Copyright terms: Public domain W3C validator