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Theorem pinq 9787
 Description: The representatives of positive integers as positive fractions. (Contributed by NM, 29-Oct-1995.) (Revised by Mario Carneiro, 6-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
pinq (𝐴N → ⟨𝐴, 1𝑜⟩ ∈ Q)

Proof of Theorem pinq
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1pi 9743 . . . 4 1𝑜N
2 opelxpi 5182 . . . 4 ((𝐴N ∧ 1𝑜N) → ⟨𝐴, 1𝑜⟩ ∈ (N × N))
31, 2mpan2 707 . . 3 (𝐴N → ⟨𝐴, 1𝑜⟩ ∈ (N × N))
4 nlt1pi 9766 . . . . . 6 ¬ (2nd𝑦) <N 1𝑜
51elexi 3244 . . . . . . . 8 1𝑜 ∈ V
6 op2ndg 7223 . . . . . . . 8 ((𝐴N ∧ 1𝑜 ∈ V) → (2nd ‘⟨𝐴, 1𝑜⟩) = 1𝑜)
75, 6mpan2 707 . . . . . . 7 (𝐴N → (2nd ‘⟨𝐴, 1𝑜⟩) = 1𝑜)
87breq2d 4697 . . . . . 6 (𝐴N → ((2nd𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩) ↔ (2nd𝑦) <N 1𝑜))
94, 8mtbiri 316 . . . . 5 (𝐴N → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩))
109a1d 25 . . . 4 (𝐴N → (⟨𝐴, 1𝑜⟩ ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩)))
1110ralrimivw 2996 . . 3 (𝐴N → ∀𝑦 ∈ (N × N)(⟨𝐴, 1𝑜⟩ ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩)))
12 breq1 4688 . . . . . 6 (𝑥 = ⟨𝐴, 1𝑜⟩ → (𝑥 ~Q 𝑦 ↔ ⟨𝐴, 1𝑜⟩ ~Q 𝑦))
13 fveq2 6229 . . . . . . . 8 (𝑥 = ⟨𝐴, 1𝑜⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 1𝑜⟩))
1413breq2d 4697 . . . . . . 7 (𝑥 = ⟨𝐴, 1𝑜⟩ → ((2nd𝑦) <N (2nd𝑥) ↔ (2nd𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩)))
1514notbid 307 . . . . . 6 (𝑥 = ⟨𝐴, 1𝑜⟩ → (¬ (2nd𝑦) <N (2nd𝑥) ↔ ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩)))
1612, 15imbi12d 333 . . . . 5 (𝑥 = ⟨𝐴, 1𝑜⟩ → ((𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥)) ↔ (⟨𝐴, 1𝑜⟩ ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩))))
1716ralbidv 3015 . . . 4 (𝑥 = ⟨𝐴, 1𝑜⟩ → (∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥)) ↔ ∀𝑦 ∈ (N × N)(⟨𝐴, 1𝑜⟩ ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩))))
1817elrab 3396 . . 3 (⟨𝐴, 1𝑜⟩ ∈ {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥))} ↔ (⟨𝐴, 1𝑜⟩ ∈ (N × N) ∧ ∀𝑦 ∈ (N × N)(⟨𝐴, 1𝑜⟩ ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩))))
193, 11, 18sylanbrc 699 . 2 (𝐴N → ⟨𝐴, 1𝑜⟩ ∈ {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥))})
20 df-nq 9772 . 2 Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥))}
2119, 20syl6eleqr 2741 1 (𝐴N → ⟨𝐴, 1𝑜⟩ ∈ Q)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1523   ∈ wcel 2030  ∀wral 2941  {crab 2945  Vcvv 3231  ⟨cop 4216   class class class wbr 4685   × cxp 5141  ‘cfv 5926  2nd c2nd 7209  1𝑜c1o 7598  Ncnpi 9704
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