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Mirrors > Home > ILE Home > Th. List > axpre-apti | GIF version |
Description: Apartness of reals is
tight. Axiom for real and complex numbers,
derived from set theory. This construction-dependent theorem should not
be referenced directly; instead, use ax-pre-apti 6999.
(Contributed by Jim Kingdon, 29-Jan-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axpre-apti | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)) → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elreal 6905 | . . 3 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 6905 | . . 3 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | breq1 3767 | . . . . . 6 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 〈𝑦, 0R〉)) | |
4 | breq2 3768 | . . . . . 6 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 〈𝑦, 0R〉 <ℝ 𝐴)) | |
5 | 3, 4 | orbi12d 707 | . . . . 5 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ (𝐴 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴))) |
6 | 5 | notbid 592 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (¬ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ ¬ (𝐴 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴))) |
7 | eqeq1 2046 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ 𝐴 = 〈𝑦, 0R〉)) | |
8 | 6, 7 | imbi12d 223 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((¬ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) → 〈𝑥, 0R〉 = 〈𝑦, 0R〉) ↔ (¬ (𝐴 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴) → 𝐴 = 〈𝑦, 0R〉))) |
9 | breq2 3768 | . . . . . 6 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 𝐵)) | |
10 | breq1 3767 | . . . . . 6 ⊢ (〈𝑦, 0R〉 = 𝐵 → (〈𝑦, 0R〉 <ℝ 𝐴 ↔ 𝐵 <ℝ 𝐴)) | |
11 | 9, 10 | orbi12d 707 | . . . . 5 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((𝐴 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴) ↔ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴))) |
12 | 11 | notbid 592 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (¬ (𝐴 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴) ↔ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴))) |
13 | eqeq2 2049 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 = 〈𝑦, 0R〉 ↔ 𝐴 = 𝐵)) | |
14 | 12, 13 | imbi12d 223 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((¬ (𝐴 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴) → 𝐴 = 〈𝑦, 0R〉) ↔ (¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴) → 𝐴 = 𝐵))) |
15 | aptisr 6863 | . . . . 5 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ ¬ (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥)) → 𝑥 = 𝑦) | |
16 | 15 | 3expia 1106 | . . . 4 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (¬ (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥) → 𝑥 = 𝑦)) |
17 | ltresr 6915 | . . . . . 6 ⊢ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝑥 <R 𝑦) | |
18 | ltresr 6915 | . . . . . 6 ⊢ (〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 𝑦 <R 𝑥) | |
19 | 17, 18 | orbi12i 681 | . . . . 5 ⊢ ((〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥)) |
20 | 19 | notbii 594 | . . . 4 ⊢ (¬ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ ¬ (𝑥 <R 𝑦 ∨ 𝑦 <R 𝑥)) |
21 | vex 2560 | . . . . 5 ⊢ 𝑥 ∈ V | |
22 | 21 | eqresr 6912 | . . . 4 ⊢ (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ 𝑥 = 𝑦) |
23 | 16, 20, 22 | 3imtr4g 194 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (¬ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) → 〈𝑥, 0R〉 = 〈𝑦, 0R〉)) |
24 | 1, 2, 8, 14, 23 | 2gencl 2587 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴) → 𝐴 = 𝐵)) |
25 | 24 | 3impia 1101 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ (𝐴 <ℝ 𝐵 ∨ 𝐵 <ℝ 𝐴)) → 𝐴 = 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 97 ∨ wo 629 ∧ w3a 885 = wceq 1243 ∈ wcel 1393 〈cop 3378 class class class wbr 3764 Rcnr 6395 0Rc0r 6396 <R cltr 6401 ℝcr 6888 <ℝ cltrr 6893 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-dc 743 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-eprel 4026 df-id 4030 df-po 4033 df-iso 4034 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 df-recs 5920 df-irdg 5957 df-1o 6001 df-2o 6002 df-oadd 6005 df-omul 6006 df-er 6106 df-ec 6108 df-qs 6112 df-ni 6402 df-pli 6403 df-mi 6404 df-lti 6405 df-plpq 6442 df-mpq 6443 df-enq 6445 df-nqqs 6446 df-plqqs 6447 df-mqqs 6448 df-1nqqs 6449 df-rq 6450 df-ltnqqs 6451 df-enq0 6522 df-nq0 6523 df-0nq0 6524 df-plq0 6525 df-mq0 6526 df-inp 6564 df-i1p 6565 df-iplp 6566 df-iltp 6568 df-enr 6811 df-nr 6812 df-ltr 6815 df-0r 6816 df-r 6899 df-lt 6902 |
This theorem is referenced by: (None) |
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