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Theorem pinq 10919
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 5151 . . . . 5 (𝑥 = ⟨𝐴, 1o⟩ → (𝑥 ~Q 𝑦 ↔ ⟨𝐴, 1o⟩ ~Q 𝑦))
2 fveq2 6889 . . . . . . 7 (𝑥 = ⟨𝐴, 1o⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 1o⟩))
32breq2d 5160 . . . . . 6 (𝑥 = ⟨𝐴, 1o⟩ → ((2nd𝑦) <N (2nd𝑥) ↔ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
43notbid 318 . . . . 5 (𝑥 = ⟨𝐴, 1o⟩ → (¬ (2nd𝑦) <N (2nd𝑥) ↔ ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
51, 4imbi12d 345 . . . 4 (𝑥 = ⟨𝐴, 1o⟩ → ((𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥)) ↔ (⟨𝐴, 1o⟩ ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩))))
65ralbidv 3178 . . 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 10875 . . . 4 1oN
8 opelxpi 5713 . . . 4 ((𝐴N ∧ 1oN) → ⟨𝐴, 1o⟩ ∈ (N × N))
97, 8mpan2 690 . . 3 (𝐴N → ⟨𝐴, 1o⟩ ∈ (N × N))
10 nlt1pi 10898 . . . . . 6 ¬ (2nd𝑦) <N 1o
11 1oex 8473 . . . . . . . 8 1o ∈ V
12 op2ndg 7985 . . . . . . . 8 ((𝐴N ∧ 1o ∈ V) → (2nd ‘⟨𝐴, 1o⟩) = 1o)
1311, 12mpan2 690 . . . . . . 7 (𝐴N → (2nd ‘⟨𝐴, 1o⟩) = 1o)
1413breq2d 5160 . . . . . 6 (𝐴N → ((2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩) ↔ (2nd𝑦) <N 1o))
1510, 14mtbiri 327 . . . . 5 (𝐴N → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩))
1615a1d 25 . . . 4 (𝐴N → (⟨𝐴, 1o⟩ ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
1716ralrimivw 3151 . . 3 (𝐴N → ∀𝑦 ∈ (N × N)(⟨𝐴, 1o⟩ ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
186, 9, 17elrabd 3685 . 2 (𝐴N → ⟨𝐴, 1o⟩ ∈ {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥))})
19 df-nq 10904 . 2 Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥))}
2018, 19eleqtrrdi 2845 1 (𝐴N → ⟨𝐴, 1o⟩ ∈ Q)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1542  wcel 2107  wral 3062  {crab 3433  Vcvv 3475  cop 4634   class class class wbr 5148   × cxp 5674  cfv 6541  2nd c2nd 7971  1oc1o 8456  Ncnpi 10836   <N clti 10839   ~Q ceq 10843  Qcnq 10844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fv 6549  df-om 7853  df-2nd 7973  df-1o 8463  df-ni 10864  df-lti 10867  df-nq 10904
This theorem is referenced by:  1nq  10920  archnq  10972  prlem934  11025
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