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Mirrors > Home > MPE Home > Th. List > pinq | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
pinq | ⊢ (𝐴 ∈ N → ⟨𝐴, 1o⟩ ∈ Q) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 5152 | . . . . 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 5161 | . . . . . 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 345 | . . . 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 3178 | . . 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 10878 | . . . 4 ⊢ 1o ∈ N | |
8 | opelxpi 5714 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N) → ⟨𝐴, 1o⟩ ∈ (N × N)) | |
9 | 7, 8 | mpan2 690 | . . 3 ⊢ (𝐴 ∈ N → ⟨𝐴, 1o⟩ ∈ (N × N)) |
10 | nlt1pi 10901 | . . . . . 6 ⊢ ¬ (2nd ‘𝑦) <N 1o | |
11 | 1oex 8476 | . . . . . . . 8 ⊢ 1o ∈ V | |
12 | op2ndg 7988 | . . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 1o ∈ V) → (2nd ‘⟨𝐴, 1o⟩) = 1o) | |
13 | 11, 12 | mpan2 690 | . . . . . . 7 ⊢ (𝐴 ∈ N → (2nd ‘⟨𝐴, 1o⟩) = 1o) |
14 | 13 | breq2d 5161 | . . . . . 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 3151 | . . 3 ⊢ (𝐴 ∈ N → ∀𝑦 ∈ (N × N)(⟨𝐴, 1o⟩ ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘⟨𝐴, 1o⟩))) |
18 | 6, 9, 17 | elrabd 3686 | . 2 ⊢ (𝐴 ∈ N → ⟨𝐴, 1o⟩ ∈ {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))}) |
19 | df-nq 10907 | . 2 ⊢ Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))} | |
20 | 18, 19 | eleqtrrdi 2845 | 1 ⊢ (𝐴 ∈ N → ⟨𝐴, 1o⟩ ∈ Q) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1542 ∈ wcel 2107 ∀wral 3062 {crab 3433 Vcvv 3475 ⟨cop 4635 class class class wbr 5149 × cxp 5675 ‘cfv 6544 2nd c2nd 7974 1oc1o 8459 Ncnpi 10839 <N clti 10842 ~Q ceq 10846 Qcnq 10847 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fv 6552 df-om 7856 df-2nd 7976 df-1o 8466 df-ni 10867 df-lti 10870 df-nq 10907 |
This theorem is referenced by: 1nq 10923 archnq 10975 prlem934 11028 |
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