Theorem pinq 9628

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		1pi 9584	. . . 4 ⊢ 1𝑜 ∈ N
2		opelxpi 5072	. . . 4 ⊢ ((𝐴 ∈ N ∧ 1𝑜 ∈ N) → ⟨𝐴, 1𝑜⟩ ∈ (N × N))
3	1, 2	mpan2 703	. . 3 ⊢ (𝐴 ∈ N → ⟨𝐴, 1𝑜⟩ ∈ (N × N))
4		nlt1pi 9607	. . . . . 6 ⊢ ¬ (2nd ‘𝑦) <N 1𝑜
5	1	elexi 3186	. . . . . . . 8 ⊢ 1𝑜 ∈ V
6		op2ndg 7072	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 1𝑜 ∈ V) → (2nd ‘⟨𝐴, 1𝑜⟩) = 1𝑜)
7	5, 6	mpan2 703	. . . . . . 7 ⊢ (𝐴 ∈ N → (2nd ‘⟨𝐴, 1𝑜⟩) = 1𝑜)
8	7	breq2d 4595	. . . . . 6 ⊢ (𝐴 ∈ N → ((2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩) ↔ (2nd ‘𝑦) <N 1𝑜))
9	4, 8	mtbiri 316	. . . . 5 ⊢ (𝐴 ∈ N → ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩))
10	9	a1d 25	. . . 4 ⊢ (𝐴 ∈ N → (⟨𝐴, 1𝑜⟩ ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩)))
11	10	ralrimivw 2950	. . 3 ⊢ (𝐴 ∈ N → ∀𝑦 ∈ (N × N)(⟨𝐴, 1𝑜⟩ ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩)))
12		breq1 4586	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 1𝑜⟩ → (𝑥 ~Q 𝑦 ↔ ⟨𝐴, 1𝑜⟩ ~Q 𝑦))
13		fveq2 6103	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 1𝑜⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 1𝑜⟩))
14	13	breq2d 4595	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 1𝑜⟩ → ((2nd ‘𝑦) <N (2nd ‘𝑥) ↔ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩)))
15	14	notbid 307	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 1𝑜⟩ → (¬ (2nd ‘𝑦) <N (2nd ‘𝑥) ↔ ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩)))
16	12, 15	imbi12d 333	. . . . 5 ⊢ (𝑥 = ⟨𝐴, 1𝑜⟩ → ((𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥)) ↔ (⟨𝐴, 1𝑜⟩ ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩))))
17	16	ralbidv 2969	. . . 4 ⊢ (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⟩))))
18	17	elrab 3331	. . 3 ⊢ (⟨𝐴, 1𝑜⟩ ∈ {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))} ↔ (⟨𝐴, 1𝑜⟩ ∈ (N × N) ∧ ∀𝑦 ∈ (N × N)(⟨𝐴, 1𝑜⟩ ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩))))
19	3, 11, 18	sylanbrc 695	. 2 ⊢ (𝐴 ∈ N → ⟨𝐴, 1𝑜⟩ ∈ {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))})
20		df-nq 9613	. 2 ⊢ Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))}
21	19, 20	syl6eleqr 2699	1 ⊢ (𝐴 ∈ N → ⟨𝐴, 1𝑜⟩ ∈ Q)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173  ⟨cop 4131   class class class wbr 4583   × cxp 5036  ‘cfv 5804  2^nd c2nd 7058  1_𝑜c1o 7440  Ncnpi 9545   <_N clti 9548   ~_Q ceq 9552  Qcnq 9553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fv 5812  df-om 6958  df-2nd 7060  df-1o 7447  df-ni 9573  df-lti 9576  df-nq 9613

This theorem is referenced by:  1nq  9629  archnq  9681  prlem934  9734