Theorem pinq 9749

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 9705	. . . 4 ⊢ 1𝑜 ∈ N
2		opelxpi 5148	. . . 4 ⊢ ((𝐴 ∈ N ∧ 1𝑜 ∈ N) → ⟨𝐴, 1𝑜⟩ ∈ (N × N))
3	1, 2	mpan2 707	. . 3 ⊢ (𝐴 ∈ N → ⟨𝐴, 1𝑜⟩ ∈ (N × N))
4		nlt1pi 9728	. . . . . 6 ⊢ ¬ (2nd ‘𝑦) <N 1𝑜
5	1	elexi 3213	. . . . . . . 8 ⊢ 1𝑜 ∈ V
6		op2ndg 7181	. . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 1𝑜 ∈ V) → (2nd ‘⟨𝐴, 1𝑜⟩) = 1𝑜)
7	5, 6	mpan2 707	. . . . . . 7 ⊢ (𝐴 ∈ N → (2nd ‘⟨𝐴, 1𝑜⟩) = 1𝑜)
8	7	breq2d 4665	. . . . . 6 ⊢ (𝐴 ∈ N → ((2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩) ↔ (2nd ‘𝑦) <N 1𝑜))
9	4, 8	mtbiri 317	. . . . 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 2967	. . 3 ⊢ (𝐴 ∈ N → ∀𝑦 ∈ (N × N)(⟨𝐴, 1𝑜⟩ ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩)))
12		breq1 4656	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 1𝑜⟩ → (𝑥 ~Q 𝑦 ↔ ⟨𝐴, 1𝑜⟩ ~Q 𝑦))
13		fveq2 6191	. . . . . . . 8 ⊢ (𝑥 = ⟨𝐴, 1𝑜⟩ → (2nd ‘𝑥) = (2nd ‘⟨𝐴, 1𝑜⟩))
14	13	breq2d 4665	. . . . . . 7 ⊢ (𝑥 = ⟨𝐴, 1𝑜⟩ → ((2nd ‘𝑦) <N (2nd ‘𝑥) ↔ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩)))
15	14	notbid 308	. . . . . 6 ⊢ (𝑥 = ⟨𝐴, 1𝑜⟩ → (¬ (2nd ‘𝑦) <N (2nd ‘𝑥) ↔ ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1𝑜⟩)))
16	12, 15	imbi12d 334	. . . . 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 2986	. . . 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 3363	. . 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 698	. 2 ⊢ (𝐴 ∈ N → ⟨𝐴, 1𝑜⟩ ∈ {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))})
20		df-nq 9734	. 2 ⊢ Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))}
21	19, 20	syl6eleqr 2712	1 ⊢ (𝐴 ∈ N → ⟨𝐴, 1𝑜⟩ ∈ Q)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1483   ∈ wcel 1990  ∀wral 2912  {crab 2916  Vcvv 3200  ⟨cop 4183   class class class wbr 4653   × cxp 5112  ‘cfv 5888  2^nd c2nd 7167  1_𝑜c1o 7553  Ncnpi 9666   <_N clti 9669   ~_Q ceq 9673  Qcnq 9674

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fv 5896  df-om 7066  df-2nd 7169  df-1o 7560  df-ni 9694  df-lti 9697  df-nq 9734

This theorem is referenced by:  1nq  9750  archnq  9802  prlem934  9855