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Theorem pinq 10337
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 5060 . . . . 5 (𝑥 = ⟨𝐴, 1o⟩ → (𝑥 ~Q 𝑦 ↔ ⟨𝐴, 1o⟩ ~Q 𝑦))
2 fveq2 6663 . . . . . . 7 (𝑥 = ⟨𝐴, 1o⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 1o⟩))
32breq2d 5069 . . . . . 6 (𝑥 = ⟨𝐴, 1o⟩ → ((2nd𝑦) <N (2nd𝑥) ↔ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
43notbid 319 . . . . 5 (𝑥 = ⟨𝐴, 1o⟩ → (¬ (2nd𝑦) <N (2nd𝑥) ↔ ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
51, 4imbi12d 346 . . . 4 (𝑥 = ⟨𝐴, 1o⟩ → ((𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥)) ↔ (⟨𝐴, 1o⟩ ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩))))
65ralbidv 3194 . . 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 10293 . . . 4 1oN
8 opelxpi 5585 . . . 4 ((𝐴N ∧ 1oN) → ⟨𝐴, 1o⟩ ∈ (N × N))
97, 8mpan2 687 . . 3 (𝐴N → ⟨𝐴, 1o⟩ ∈ (N × N))
10 nlt1pi 10316 . . . . . 6 ¬ (2nd𝑦) <N 1o
11 1oex 8099 . . . . . . . 8 1o ∈ V
12 op2ndg 7691 . . . . . . . 8 ((𝐴N ∧ 1o ∈ V) → (2nd ‘⟨𝐴, 1o⟩) = 1o)
1311, 12mpan2 687 . . . . . . 7 (𝐴N → (2nd ‘⟨𝐴, 1o⟩) = 1o)
1413breq2d 5069 . . . . . 6 (𝐴N → ((2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩) ↔ (2nd𝑦) <N 1o))
1510, 14mtbiri 328 . . . . 5 (𝐴N → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩))
1615a1d 25 . . . 4 (𝐴N → (⟨𝐴, 1o⟩ ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
1716ralrimivw 3180 . . 3 (𝐴N → ∀𝑦 ∈ (N × N)(⟨𝐴, 1o⟩ ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
186, 9, 17elrabd 3679 . 2 (𝐴N → ⟨𝐴, 1o⟩ ∈ {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥))})
19 df-nq 10322 . 2 Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd𝑦) <N (2nd𝑥))}
2018, 19eleqtrrdi 2921 1 (𝐴N → ⟨𝐴, 1o⟩ ∈ Q)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492  cop 4563   class class class wbr 5057   × cxp 5546  cfv 6348  2nd c2nd 7677  1oc1o 8084  Ncnpi 10254   <N clti 10257   ~Q ceq 10261  Qcnq 10262
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fv 6356  df-om 7570  df-2nd 7679  df-1o 8091  df-ni 10282  df-lti 10285  df-nq 10322
This theorem is referenced by:  1nq  10338  archnq  10390  prlem934  10443
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