Theorem pinq 10965

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 5151	. . . . 5 ⊢ (𝑥 = ⟨𝐴, 1o⟩ → (𝑥 ~Q 𝑦 ↔ ⟨𝐴, 1o⟩ ~Q 𝑦))
2		fveq2 6907	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 1o⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 1o⟩))
3	2	breq2d 5160	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 1o⟩ → ((2nd ‘𝑦) <N (2nd ‘𝑥) ↔ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
4	3	notbid 318	. . . . 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 3176	. . 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 10921	. . . 4 ⊢ 1o ∈ N
8		opelxpi 5726	. . . 4 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N) → ⟨𝐴, 1o⟩ ∈ (N × N))
9	7, 8	mpan2 691	. . 3 ⊢ (𝐴 ∈ N → ⟨𝐴, 1o⟩ ∈ (N × N))
10		nlt1pi 10944	. . . . . 6 ⊢ ¬ (2nd ‘𝑦) <N 1o
11		1oex 8515	. . . . . . . 8 ⊢ 1o ∈ V
12		op2ndg 8026	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 1o ∈ V) → (2nd ‘⟨𝐴, 1o⟩) = 1o)
13	11, 12	mpan2 691	. . . . . . 7 ⊢ (𝐴 ∈ N → (2nd ‘⟨𝐴, 1o⟩) = 1o)
14	13	breq2d 5160	. . . . . 6 ⊢ (𝐴 ∈ N → ((2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1o⟩) ↔ (2nd ‘𝑦) <N 1o))
15	10, 14	mtbiri 327	. . . . 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 3148	. . 3 ⊢ (𝐴 ∈ N → ∀𝑦 ∈ (N × N)(⟨𝐴, 1o⟩ ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
18	6, 9, 17	elrabd 3697	. 2 ⊢ (𝐴 ∈ N → ⟨𝐴, 1o⟩ ∈ {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))})
19		df-nq 10950	. 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 1537   ∈ wcel 2106  ∀wral 3059  {crab 3433  Vcvv 3478  ⟨cop 4637   class class class wbr 5148   × cxp 5687  ‘cfv 6563  2^nd c2nd 8012  1_oc1o 8498  Ncnpi 10882   <_N clti 10885   ~_Q ceq 10889  Qcnq 10890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fv 6571  df-om 7888  df-2nd 8014  df-1o 8505  df-ni 10910  df-lti 10913  df-nq 10950

This theorem is referenced by:  1nq  10966  archnq  11018  prlem934  11071