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Theorem nqprm 7373
 Description: A cut produced from a rational is inhabited. Lemma for nqprlu 7378. (Contributed by Jim Kingdon, 8-Dec-2019.)
Assertion
Ref Expression
nqprm (𝐴Q → (∃𝑞Q 𝑞 ∈ {𝑥𝑥 <Q 𝐴} ∧ ∃𝑟Q 𝑟 ∈ {𝑥𝐴 <Q 𝑥}))
Distinct variable group:   𝑥,𝐴,𝑟,𝑞

Proof of Theorem nqprm
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 nsmallnqq 7243 . . 3 (𝐴Q → ∃𝑞Q 𝑞 <Q 𝐴)
2 vex 2692 . . . . 5 𝑞 ∈ V
3 breq1 3939 . . . . 5 (𝑥 = 𝑞 → (𝑥 <Q 𝐴𝑞 <Q 𝐴))
42, 3elab 2831 . . . 4 (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ↔ 𝑞 <Q 𝐴)
54rexbii 2445 . . 3 (∃𝑞Q 𝑞 ∈ {𝑥𝑥 <Q 𝐴} ↔ ∃𝑞Q 𝑞 <Q 𝐴)
61, 5sylibr 133 . 2 (𝐴Q → ∃𝑞Q 𝑞 ∈ {𝑥𝑥 <Q 𝐴})
7 archnqq 7248 . . . . 5 (𝐴Q → ∃𝑛N 𝐴 <Q [⟨𝑛, 1o⟩] ~Q )
8 df-rex 2423 . . . . 5 (∃𝑛N 𝐴 <Q [⟨𝑛, 1o⟩] ~Q ↔ ∃𝑛(𝑛N𝐴 <Q [⟨𝑛, 1o⟩] ~Q ))
97, 8sylib 121 . . . 4 (𝐴Q → ∃𝑛(𝑛N𝐴 <Q [⟨𝑛, 1o⟩] ~Q ))
10 1pi 7146 . . . . . . . 8 1oN
11 opelxpi 4578 . . . . . . . . 9 ((𝑛N ∧ 1oN) → ⟨𝑛, 1o⟩ ∈ (N × N))
12 enqex 7191 . . . . . . . . . 10 ~Q ∈ V
1312ecelqsi 6490 . . . . . . . . 9 (⟨𝑛, 1o⟩ ∈ (N × N) → [⟨𝑛, 1o⟩] ~Q ∈ ((N × N) / ~Q ))
1411, 13syl 14 . . . . . . . 8 ((𝑛N ∧ 1oN) → [⟨𝑛, 1o⟩] ~Q ∈ ((N × N) / ~Q ))
1510, 14mpan2 422 . . . . . . 7 (𝑛N → [⟨𝑛, 1o⟩] ~Q ∈ ((N × N) / ~Q ))
16 df-nqqs 7179 . . . . . . 7 Q = ((N × N) / ~Q )
1715, 16eleqtrrdi 2234 . . . . . 6 (𝑛N → [⟨𝑛, 1o⟩] ~QQ)
18 breq2 3940 . . . . . . 7 (𝑟 = [⟨𝑛, 1o⟩] ~Q → (𝐴 <Q 𝑟𝐴 <Q [⟨𝑛, 1o⟩] ~Q ))
1918rspcev 2792 . . . . . 6 (([⟨𝑛, 1o⟩] ~QQ𝐴 <Q [⟨𝑛, 1o⟩] ~Q ) → ∃𝑟Q 𝐴 <Q 𝑟)
2017, 19sylan 281 . . . . 5 ((𝑛N𝐴 <Q [⟨𝑛, 1o⟩] ~Q ) → ∃𝑟Q 𝐴 <Q 𝑟)
2120exlimiv 1578 . . . 4 (∃𝑛(𝑛N𝐴 <Q [⟨𝑛, 1o⟩] ~Q ) → ∃𝑟Q 𝐴 <Q 𝑟)
229, 21syl 14 . . 3 (𝐴Q → ∃𝑟Q 𝐴 <Q 𝑟)
23 vex 2692 . . . . 5 𝑟 ∈ V
24 breq2 3940 . . . . 5 (𝑥 = 𝑟 → (𝐴 <Q 𝑥𝐴 <Q 𝑟))
2523, 24elab 2831 . . . 4 (𝑟 ∈ {𝑥𝐴 <Q 𝑥} ↔ 𝐴 <Q 𝑟)
2625rexbii 2445 . . 3 (∃𝑟Q 𝑟 ∈ {𝑥𝐴 <Q 𝑥} ↔ ∃𝑟Q 𝐴 <Q 𝑟)
2722, 26sylibr 133 . 2 (𝐴Q → ∃𝑟Q 𝑟 ∈ {𝑥𝐴 <Q 𝑥})
286, 27jca 304 1 (𝐴Q → (∃𝑞Q 𝑞 ∈ {𝑥𝑥 <Q 𝐴} ∧ ∃𝑟Q 𝑟 ∈ {𝑥𝐴 <Q 𝑥}))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103  ∃wex 1469   ∈ wcel 1481  {cab 2126  ∃wrex 2418  ⟨cop 3534   class class class wbr 3936   × cxp 4544  1oc1o 6313  [cec 6434   / cqs 6435  Ncnpi 7103   ~Q ceq 7110  Qcnq 7111
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