Theorem pinq 10945

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 5146	. . . . 5 ⊢ (𝑥 = ⟨𝐴, 1o⟩ → (𝑥 ~Q 𝑦 ↔ ⟨𝐴, 1o⟩ ~Q 𝑦))
2		fveq2 6892	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 1o⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 1o⟩))
3	2	breq2d 5155	. . . . . 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 3173	. . 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 10901	. . . 4 ⊢ 1o ∈ N
8		opelxpi 5710	. . . 4 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N) → ⟨𝐴, 1o⟩ ∈ (N × N))
9	7, 8	mpan2 690	. . 3 ⊢ (𝐴 ∈ N → ⟨𝐴, 1o⟩ ∈ (N × N))
10		nlt1pi 10924	. . . . . 6 ⊢ ¬ (2nd ‘𝑦) <N 1o
11		1oex 8491	. . . . . . . 8 ⊢ 1o ∈ V
12		op2ndg 8001	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 1o ∈ V) → (2nd ‘⟨𝐴, 1o⟩) = 1o)
13	11, 12	mpan2 690	. . . . . . 7 ⊢ (𝐴 ∈ N → (2nd ‘⟨𝐴, 1o⟩) = 1o)
14	13	breq2d 5155	. . . . . 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 3146	. . 3 ⊢ (𝐴 ∈ N → ∀𝑦 ∈ (N × N)(⟨𝐴, 1o⟩ ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1o⟩)))
18	6, 9, 17	elrabd 3683	. 2 ⊢ (𝐴 ∈ N → ⟨𝐴, 1o⟩ ∈ {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))})
19		df-nq 10930	. 2 ⊢ Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))}
20	18, 19	eleqtrrdi 2840	1 ⊢ (𝐴 ∈ N → ⟨𝐴, 1o⟩ ∈ Q)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1534   ∈ wcel 2099  ∀wral 3057  {crab 3428  Vcvv 3470  ⟨cop 4631   class class class wbr 5143   × cxp 5671  ‘cfv 6543  2^nd c2nd 7987  1_oc1o 8474  Ncnpi 10862   <_N clti 10865   ~_Q ceq 10869  Qcnq 10870

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fv 6551  df-om 7866  df-2nd 7989  df-1o 8481  df-ni 10890  df-lti 10893  df-nq 10930

This theorem is referenced by:  1nq  10946  archnq  10998  prlem934  11051

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