MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pinq Structured version   Visualization version   GIF version

Theorem pinq 10922
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 5152 . . . . 5 (𝑥 = ⟨𝐴, 1o⟩ → (𝑥 ~Q 𝑦 ↔ ⟨𝐴, 1o⟩ ~Q 𝑦))
2 fveq2 6892 . . . . . . 7 (𝑥 = ⟨𝐴, 1o⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 1o⟩))
32breq2d 5161 . . . . . 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 10878 . . . 4 1oN
8 opelxpi 5714 . . . 4 ((𝐴N ∧ 1oN) → ⟨𝐴, 1o⟩ ∈ (N × N))
97, 8mpan2 690 . . 3 (𝐴N → ⟨𝐴, 1o⟩ ∈ (N × N))
10 nlt1pi 10901 . . . . . 6 ¬ (2nd𝑦) <N 1o
11 1oex 8476 . . . . . . . 8 1o ∈ V
12 op2ndg 7988 . . . . . . . 8 ((𝐴N ∧ 1o ∈ V) → (2nd ‘⟨𝐴, 1o⟩) = 1o)
1311, 12mpan2 690 . . . . . . 7 (𝐴N → (2nd ‘⟨𝐴, 1o⟩) = 1o)
1413breq2d 5161 . . . . . 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 3686 . 2 (𝐴N → ⟨𝐴, 1o⟩ ∈ {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥))})
19 df-nq 10907 . 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 4635   class class class wbr 5149   × cxp 5675  cfv 6544  2nd c2nd 7974  1oc1o 8459  Ncnpi 10839   <N clti 10842   ~Q ceq 10846  Qcnq 10847
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 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
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 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fv 6552  df-om 7856  df-2nd 7976  df-1o 8466  df-ni 10867  df-lti 10870  df-nq 10907
This theorem is referenced by:  1nq  10923  archnq  10975  prlem934  11028
  Copyright terms: Public domain W3C validator