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Mirrors > Home > ILE Home > Th. List > aptipr | GIF version |
Description: Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.) |
Ref | Expression |
---|---|
aptipr | ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 939 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐴 ∈ P) | |
2 | simp2 940 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐵 ∈ P) | |
3 | ioran 702 | . . . . . . 7 ⊢ (¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴) ↔ (¬ 𝐴<P 𝐵 ∧ ¬ 𝐵<P 𝐴)) | |
4 | 3 | biimpi 118 | . . . . . 6 ⊢ (¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴) → (¬ 𝐴<P 𝐵 ∧ ¬ 𝐵<P 𝐴)) |
5 | 4 | 3ad2ant3 962 | . . . . 5 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (¬ 𝐴<P 𝐵 ∧ ¬ 𝐵<P 𝐴)) |
6 | 5 | simprd 112 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → ¬ 𝐵<P 𝐴) |
7 | aptiprleml 6961 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (1st ‘𝐴) ⊆ (1st ‘𝐵)) | |
8 | 1, 2, 6, 7 | syl3anc 1170 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (1st ‘𝐴) ⊆ (1st ‘𝐵)) |
9 | 5 | simpld 110 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → ¬ 𝐴<P 𝐵) |
10 | aptiprleml 6961 | . . . 4 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P ∧ ¬ 𝐴<P 𝐵) → (1st ‘𝐵) ⊆ (1st ‘𝐴)) | |
11 | 2, 1, 9, 10 | syl3anc 1170 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (1st ‘𝐵) ⊆ (1st ‘𝐴)) |
12 | 8, 11 | eqssd 3025 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (1st ‘𝐴) = (1st ‘𝐵)) |
13 | aptiprlemu 6962 | . . . 4 ⊢ ((𝐵 ∈ P ∧ 𝐴 ∈ P ∧ ¬ 𝐴<P 𝐵) → (2nd ‘𝐴) ⊆ (2nd ‘𝐵)) | |
14 | 2, 1, 9, 13 | syl3anc 1170 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (2nd ‘𝐴) ⊆ (2nd ‘𝐵)) |
15 | aptiprlemu 6962 | . . . 4 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ 𝐵<P 𝐴) → (2nd ‘𝐵) ⊆ (2nd ‘𝐴)) | |
16 | 1, 2, 6, 15 | syl3anc 1170 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (2nd ‘𝐵) ⊆ (2nd ‘𝐴)) |
17 | 14, 16 | eqssd 3025 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (2nd ‘𝐴) = (2nd ‘𝐵)) |
18 | preqlu 6794 | . . 3 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P) → (𝐴 = 𝐵 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)))) | |
19 | 18 | 3adant3 959 | . 2 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → (𝐴 = 𝐵 ↔ ((1st ‘𝐴) = (1st ‘𝐵) ∧ (2nd ‘𝐴) = (2nd ‘𝐵)))) |
20 | 12, 17, 19 | mpbir2and 886 | 1 ⊢ ((𝐴 ∈ P ∧ 𝐵 ∈ P ∧ ¬ (𝐴<P 𝐵 ∨ 𝐵<P 𝐴)) → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ↔ wb 103 ∨ wo 662 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 ⊆ wss 2982 class class class wbr 3805 ‘cfv 4952 1st c1st 5817 2nd c2nd 5818 Pcnp 6613 <P cltp 6617 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3913 ax-sep 3916 ax-nul 3924 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 ax-iinf 4357 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2612 df-sbc 2825 df-csb 2918 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-nul 3268 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-int 3657 df-iun 3700 df-br 3806 df-opab 3860 df-mpt 3861 df-tr 3896 df-eprel 4072 df-id 4076 df-po 4079 df-iso 4080 df-iord 4149 df-on 4151 df-suc 4154 df-iom 4360 df-xp 4397 df-rel 4398 df-cnv 4399 df-co 4400 df-dm 4401 df-rn 4402 df-res 4403 df-ima 4404 df-iota 4917 df-fun 4954 df-fn 4955 df-f 4956 df-f1 4957 df-fo 4958 df-f1o 4959 df-fv 4960 df-ov 5567 df-oprab 5568 df-mpt2 5569 df-1st 5819 df-2nd 5820 df-recs 5975 df-irdg 6040 df-1o 6086 df-2o 6087 df-oadd 6090 df-omul 6091 df-er 6194 df-ec 6196 df-qs 6200 df-ni 6626 df-pli 6627 df-mi 6628 df-lti 6629 df-plpq 6666 df-mpq 6667 df-enq 6669 df-nqqs 6670 df-plqqs 6671 df-mqqs 6672 df-1nqqs 6673 df-rq 6674 df-ltnqqs 6675 df-enq0 6746 df-nq0 6747 df-0nq0 6748 df-plq0 6749 df-mq0 6750 df-inp 6788 df-iltp 6792 |
This theorem is referenced by: aptisr 7087 |
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