Theorem pinq 10614

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 5073	. . . . 5 ⊢ (𝑥 = ⟨𝐴, 1o⟩ → (𝑥 ~Q 𝑦 ↔ ⟨𝐴, 1o⟩ ~Q 𝑦))
2		fveq2 6756	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 1o⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 1o⟩))
3	2	breq2d 5082	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 1o⟩ → ((2nd ‘𝑦) <N (2nd ‘𝑥) ↔ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
4	3	notbid 317	. . . . 5 ⊢ (𝑥 = ⟨𝐴, 1o⟩ → (¬ (2nd ‘𝑦) <N (2nd ‘𝑥) ↔ ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
5	1, 4	imbi12d 344	. . . 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 3120	. . 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 10570	. . . 4 ⊢ 1o ∈ N
8		opelxpi 5617	. . . 4 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N) → ⟨𝐴, 1o⟩ ∈ (N × N))
9	7, 8	mpan2 687	. . 3 ⊢ (𝐴 ∈ N → ⟨𝐴, 1o⟩ ∈ (N × N))
10		nlt1pi 10593	. . . . . 6 ⊢ ¬ (2nd ‘𝑦) <N 1o
11		1oex 8280	. . . . . . . 8 ⊢ 1o ∈ V
12		op2ndg 7817	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 1o ∈ V) → (2nd ‘⟨𝐴, 1o⟩) = 1o)
13	11, 12	mpan2 687	. . . . . . 7 ⊢ (𝐴 ∈ N → (2nd ‘⟨𝐴, 1o⟩) = 1o)
14	13	breq2d 5082	. . . . . 6 ⊢ (𝐴 ∈ N → ((2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1o⟩) ↔ (2nd ‘𝑦) <N 1o))
15	10, 14	mtbiri 326	. . . . 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 3108	. . 3 ⊢ (𝐴 ∈ N → ∀𝑦 ∈ (N × N)(⟨𝐴, 1o⟩ ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
18	6, 9, 17	elrabd 3619	. 2 ⊢ (𝐴 ∈ N → ⟨𝐴, 1o⟩ ∈ {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))})
19		df-nq 10599	. 2 ⊢ Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))}
20	18, 19	eleqtrrdi 2850	1 ⊢ (𝐴 ∈ N → ⟨𝐴, 1o⟩ ∈ Q)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ∈ wcel 2108  ∀wral 3063  {crab 3067  Vcvv 3422  ⟨cop 4564   class class class wbr 5070   × cxp 5578  ‘cfv 6418  2^nd c2nd 7803  1_oc1o 8260  Ncnpi 10531   <_N clti 10534   ~_Q ceq 10538  Qcnq 10539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fv 6426  df-om 7688  df-2nd 7805  df-1o 8267  df-ni 10559  df-lti 10562  df-nq 10599

This theorem is referenced by:  1nq  10615  archnq  10667  prlem934  10720