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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 5060 | . . . . 5 ⊢ (𝑥 = 〈𝐴, 1o〉 → (𝑥 ~Q 𝑦 ↔ 〈𝐴, 1o〉 ~Q 𝑦)) | |
2 | fveq2 6663 | . . . . . . 7 ⊢ (𝑥 = 〈𝐴, 1o〉 → (2nd ‘𝑥) = (2nd ‘〈𝐴, 1o〉)) | |
3 | 2 | breq2d 5069 | . . . . . 6 ⊢ (𝑥 = 〈𝐴, 1o〉 → ((2nd ‘𝑦) <N (2nd ‘𝑥) ↔ (2nd ‘𝑦) <N (2nd ‘〈𝐴, 1o〉))) |
4 | 3 | notbid 319 | . . . . 5 ⊢ (𝑥 = 〈𝐴, 1o〉 → (¬ (2nd ‘𝑦) <N (2nd ‘𝑥) ↔ ¬ (2nd ‘𝑦) <N (2nd ‘〈𝐴, 1o〉))) |
5 | 1, 4 | imbi12d 346 | . . . 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 3194 | . . 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 10293 | . . . 4 ⊢ 1o ∈ N | |
8 | opelxpi 5585 | . . . 4 ⊢ ((𝐴 ∈ N ∧ 1o ∈ N) → 〈𝐴, 1o〉 ∈ (N × N)) | |
9 | 7, 8 | mpan2 687 | . . 3 ⊢ (𝐴 ∈ N → 〈𝐴, 1o〉 ∈ (N × N)) |
10 | nlt1pi 10316 | . . . . . 6 ⊢ ¬ (2nd ‘𝑦) <N 1o | |
11 | 1oex 8099 | . . . . . . . 8 ⊢ 1o ∈ V | |
12 | op2ndg 7691 | . . . . . . . 8 ⊢ ((𝐴 ∈ N ∧ 1o ∈ V) → (2nd ‘〈𝐴, 1o〉) = 1o) | |
13 | 11, 12 | mpan2 687 | . . . . . . 7 ⊢ (𝐴 ∈ N → (2nd ‘〈𝐴, 1o〉) = 1o) |
14 | 13 | breq2d 5069 | . . . . . 6 ⊢ (𝐴 ∈ N → ((2nd ‘𝑦) <N (2nd ‘〈𝐴, 1o〉) ↔ (2nd ‘𝑦) <N 1o)) |
15 | 10, 14 | mtbiri 328 | . . . . 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 3180 | . . 3 ⊢ (𝐴 ∈ N → ∀𝑦 ∈ (N × N)(〈𝐴, 1o〉 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘〈𝐴, 1o〉))) |
18 | 6, 9, 17 | elrabd 3679 | . 2 ⊢ (𝐴 ∈ N → 〈𝐴, 1o〉 ∈ {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))}) |
19 | df-nq 10322 | . 2 ⊢ Q = {𝑥 ∈ (N × N) ∣ ∀𝑦 ∈ (N × N)(𝑥 ~Q 𝑦 → ¬ (2nd ‘𝑦) <N (2nd ‘𝑥))} | |
20 | 18, 19 | eleqtrrdi 2921 | 1 ⊢ (𝐴 ∈ N → 〈𝐴, 1o〉 ∈ Q) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1528 ∈ wcel 2105 ∀wral 3135 {crab 3139 Vcvv 3492 〈cop 4563 class class class wbr 5057 × cxp 5546 ‘cfv 6348 2nd c2nd 7677 1oc1o 8084 Ncnpi 10254 <N clti 10257 ~Q ceq 10261 Qcnq 10262 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fv 6356 df-om 7570 df-2nd 7679 df-1o 8091 df-ni 10282 df-lti 10285 df-nq 10322 |
This theorem is referenced by: 1nq 10338 archnq 10390 prlem934 10443 |
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