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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   <N clti 9707   ~Q ceq 9711  Qcnq 9712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fv 5934  df-om 7108  df-2nd 7211  df-1o 7605  df-ni 9732  df-lti 9735  df-nq 9772
This theorem is referenced by:  1nq  9788  archnq  9840  prlem934  9893
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