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Theorem nqprm 6794
 Description: A cut produced from a rational is inhabited. Lemma for nqprlu 6799. (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 6664 . . 3 (𝐴Q → ∃𝑞Q 𝑞 <Q 𝐴)
2 vex 2605 . . . . 5 𝑞 ∈ V
3 breq1 3796 . . . . 5 (𝑥 = 𝑞 → (𝑥 <Q 𝐴𝑞 <Q 𝐴))
42, 3elab 2739 . . . 4 (𝑞 ∈ {𝑥𝑥 <Q 𝐴} ↔ 𝑞 <Q 𝐴)
54rexbii 2374 . . 3 (∃𝑞Q 𝑞 ∈ {𝑥𝑥 <Q 𝐴} ↔ ∃𝑞Q 𝑞 <Q 𝐴)
61, 5sylibr 132 . 2 (𝐴Q → ∃𝑞Q 𝑞 ∈ {𝑥𝑥 <Q 𝐴})
7 archnqq 6669 . . . . 5 (𝐴Q → ∃𝑛N 𝐴 <Q [⟨𝑛, 1𝑜⟩] ~Q )
8 df-rex 2355 . . . . 5 (∃𝑛N 𝐴 <Q [⟨𝑛, 1𝑜⟩] ~Q ↔ ∃𝑛(𝑛N𝐴 <Q [⟨𝑛, 1𝑜⟩] ~Q ))
97, 8sylib 120 . . . 4 (𝐴Q → ∃𝑛(𝑛N𝐴 <Q [⟨𝑛, 1𝑜⟩] ~Q ))
10 1pi 6567 . . . . . . . 8 1𝑜N
11 opelxpi 4402 . . . . . . . . 9 ((𝑛N ∧ 1𝑜N) → ⟨𝑛, 1𝑜⟩ ∈ (N × N))
12 enqex 6612 . . . . . . . . . 10 ~Q ∈ V
1312ecelqsi 6226 . . . . . . . . 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 416 . . . . . . 7 (𝑛N → [⟨𝑛, 1𝑜⟩] ~Q ∈ ((N × N) / ~Q ))
16 df-nqqs 6600 . . . . . . 7 Q = ((N × N) / ~Q )
1715, 16syl6eleqr 2173 . . . . . 6 (𝑛N → [⟨𝑛, 1𝑜⟩] ~QQ)
18 breq2 3797 . . . . . . 7 (𝑟 = [⟨𝑛, 1𝑜⟩] ~Q → (𝐴 <Q 𝑟𝐴 <Q [⟨𝑛, 1𝑜⟩] ~Q ))
1918rspcev 2702 . . . . . 6 (([⟨𝑛, 1𝑜⟩] ~QQ𝐴 <Q [⟨𝑛, 1𝑜⟩] ~Q ) → ∃𝑟Q 𝐴 <Q 𝑟)
2017, 19sylan 277 . . . . 5 ((𝑛N𝐴 <Q [⟨𝑛, 1𝑜⟩] ~Q ) → ∃𝑟Q 𝐴 <Q 𝑟)
2120exlimiv 1530 . . . 4 (∃𝑛(𝑛N𝐴 <Q [⟨𝑛, 1𝑜⟩] ~Q ) → ∃𝑟Q 𝐴 <Q 𝑟)
229, 21syl 14 . . 3 (𝐴Q → ∃𝑟Q 𝐴 <Q 𝑟)
23 vex 2605 . . . . 5 𝑟 ∈ V
24 breq2 3797 . . . . 5 (𝑥 = 𝑟 → (𝐴 <Q 𝑥𝐴 <Q 𝑟))
2523, 24elab 2739 . . . 4 (𝑟 ∈ {𝑥𝐴 <Q 𝑥} ↔ 𝐴 <Q 𝑟)
2625rexbii 2374 . . 3 (∃𝑟Q 𝑟 ∈ {𝑥𝐴 <Q 𝑥} ↔ ∃𝑟Q 𝐴 <Q 𝑟)
2722, 26sylibr 132 . 2 (𝐴Q → ∃𝑟Q 𝑟 ∈ {𝑥𝐴 <Q 𝑥})
286, 27jca 300 1 (𝐴Q → (∃𝑞Q 𝑞 ∈ {𝑥𝑥 <Q 𝐴} ∧ ∃𝑟Q 𝑟 ∈ {𝑥𝐴 <Q 𝑥}))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 102  ∃wex 1422   ∈ wcel 1434  {cab 2068  ∃wrex 2350  ⟨cop 3409   class class class wbr 3793   × cxp 4369  1𝑜c1o 6058  [cec 6170   / cqs 6171  Ncnpi 6524   ~Q ceq 6531  Qcnq 6532
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