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Theorem pinq 10388
 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 breq1 5036 . . . . 5 (𝑥 = ⟨𝐴, 1o⟩ → (𝑥 ~Q 𝑦 ↔ ⟨𝐴, 1o⟩ ~Q 𝑦))
2 fveq2 6659 . . . . . . 7 (𝑥 = ⟨𝐴, 1o⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 1o⟩))
32breq2d 5045 . . . . . 6 (𝑥 = ⟨𝐴, 1o⟩ → ((2nd𝑦) <N (2nd𝑥) ↔ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
43notbid 322 . . . . 5 (𝑥 = ⟨𝐴, 1o⟩ → (¬ (2nd𝑦) <N (2nd𝑥) ↔ ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
51, 4imbi12d 349 . . . 4 (𝑥 = ⟨𝐴, 1o⟩ → ((𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥)) ↔ (⟨𝐴, 1o⟩ ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩))))
65ralbidv 3127 . . 3 (𝑥 = ⟨𝐴, 1o⟩ → (∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥)) ↔ ∀𝑦 ∈ (N × N)(⟨𝐴, 1o⟩ ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩))))
7 1pi 10344 . . . 4 1oN
8 opelxpi 5562 . . . 4 ((𝐴N ∧ 1oN) → ⟨𝐴, 1o⟩ ∈ (N × N))
97, 8mpan2 691 . . 3 (𝐴N → ⟨𝐴, 1o⟩ ∈ (N × N))
10 nlt1pi 10367 . . . . . 6 ¬ (2nd𝑦) <N 1o
11 1oex 8121 . . . . . . . 8 1o ∈ V
12 op2ndg 7707 . . . . . . . 8 ((𝐴N ∧ 1o ∈ V) → (2nd ‘⟨𝐴, 1o⟩) = 1o)
1311, 12mpan2 691 . . . . . . 7 (𝐴N → (2nd ‘⟨𝐴, 1o⟩) = 1o)
1413breq2d 5045 . . . . . 6 (𝐴N → ((2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩) ↔ (2nd𝑦) <N 1o))
1510, 14mtbiri 331 . . . . 5 (𝐴N → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩))
1615a1d 25 . . . 4 (𝐴N → (⟨𝐴, 1o⟩ ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
1716ralrimivw 3115 . . 3 (𝐴N → ∀𝑦 ∈ (N × N)(⟨𝐴, 1o⟩ ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
186, 9, 17elrabd 3605 . 2 (𝐴N → ⟨𝐴, 1o⟩ ∈ {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥))})
19 df-nq 10373 . 2 Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥))}
2018, 19eleqtrrdi 2864 1 (𝐴N → ⟨𝐴, 1o⟩ ∈ Q)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ∈ wcel 2112  ∀wral 3071  {crab 3075  Vcvv 3410  ⟨cop 4529   class class class wbr 5033   × cxp 5523  ‘cfv 6336  2nd c2nd 7693  1oc1o 8106  Ncnpi 10305
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