Theorem pinq 10887

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	⊢ (𝐴 ∈ N → ⟨𝐴, 1o⟩ ∈ Q)

Proof of Theorem pinq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 5105	. . . . 5 ⊢ (𝑥 = ⟨𝐴, 1o⟩ → (𝑥 ~Q 𝑦 ↔ ⟨𝐴, 1o⟩ ~Q 𝑦))
2		fveq2 6869	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 1o⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 1o⟩))
3	2	breq2d 5114	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 1o⟩ → ((2nd ‘𝑦) <N (2nd ‘𝑥) ↔ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
4	3	notbid 320	. . . . 5 ⊢ (𝑥 = ⟨𝐴, 1o⟩ → (¬ (2nd ‘𝑦) <N (2nd ‘𝑥) ↔ ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
5	1, 4	imbi12d 346	. . . 4 ⊢ (𝑥 = ⟨𝐴, 1o⟩ → ((𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥)) ↔ (⟨𝐴, 1o⟩ ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1o⟩))))
6	5	ralbidv 3187	. . 3 ⊢ (𝑥 = ⟨𝐴, 1o⟩ → (∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥)) ↔ ∀𝑦 ∈ (N × N)(⟨𝐴, 1o⟩ ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1o⟩))))
7		1pi 10843	. . . 4 ⊢ 1o ∈ N
8		opelxpi 5686	. . . 4 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N) → ⟨𝐴, 1o⟩ ∈ (N × N))
9	7, 8	mpan2 701	. . 3 ⊢ (𝐴 ∈ N → ⟨𝐴, 1o⟩ ∈ (N × N))
10		nlt1pi 10866	. . . . . 6 ⊢ ¬ (2nd ‘𝑦) <N 1o
11		1oex 8449	. . . . . . . 8 ⊢ 1o ∈ V
12		op2ndg 7985	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 1o ∈ V) → (2nd ‘⟨𝐴, 1o⟩) = 1o)
13	11, 12	mpan2 701	. . . . . . 7 ⊢ (𝐴 ∈ N → (2nd ‘⟨𝐴, 1o⟩) = 1o)
14	13	breq2d 5114	. . . . . 6 ⊢ (𝐴 ∈ N → ((2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1o⟩) ↔ (2nd ‘𝑦) <N 1o))
15	10, 14	mtbiri 329	. . . . 5 ⊢ (𝐴 ∈ N → ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1o⟩))
16	15	a1d 25	. . . 4 ⊢ (𝐴 ∈ N → (⟨𝐴, 1o⟩ ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
17	16	ralrimivw 3160	. . 3 ⊢ (𝐴 ∈ N → ∀𝑦 ∈ (N × N)(⟨𝐴, 1o⟩ ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
18	6, 9, 17	elrabd 3654	. 2 ⊢ (𝐴 ∈ N → ⟨𝐴, 1o⟩ ∈ {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))})
19		df-nq 10872	. 2 ⊢ Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))}
20	18, 19	eleqtrrdi 2875	1 ⊢ (𝐴 ∈ N → ⟨𝐴, 1o⟩ ∈ Q)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1562   ∈ wcel 2144  ∀wral 3078  {crab 3416  Vcvv 3456  ⟨cop 4590   class class class wbr 5102   × cxp 5647  ‘cfv 6523  2^nd c2nd 7971  1_oc1o 8432  Ncnpi 10804   <_N clti 10807   ~_Q ceq 10811  Qcnq 10812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-ord 6351  df-on 6352  df-lim 6353  df-suc 6354  df-iota 6479  df-fun 6525  df-fv 6531  df-om 7849  df-2nd 7973  df-1o 8439  df-ni 10832  df-lti 10835  df-nq 10872

This theorem is referenced by:  1nq  10888  archnq  10940  prlem934  10993