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Mirrors > Home > ILE Home > Th. List > axpre-ltwlin | GIF version |
Description: Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltwlin 7519. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axpre-ltwlin | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elreal 7427 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 7427 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | elreal 7427 | . 2 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R 〈𝑧, 0R〉 = 𝐶) | |
4 | breq1 3854 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 〈𝑦, 0R〉)) | |
5 | breq1 3854 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 <ℝ 〈𝑧, 0R〉 ↔ 𝐴 <ℝ 〈𝑧, 0R〉)) | |
6 | 5 | orbi1d 741 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((〈𝑥, 0R〉 <ℝ 〈𝑧, 0R〉 ∨ 〈𝑧, 0R〉 <ℝ 〈𝑦, 0R〉) ↔ (𝐴 <ℝ 〈𝑧, 0R〉 ∨ 〈𝑧, 0R〉 <ℝ 〈𝑦, 0R〉))) |
7 | 4, 6 | imbi12d 233 | . 2 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 → (〈𝑥, 0R〉 <ℝ 〈𝑧, 0R〉 ∨ 〈𝑧, 0R〉 <ℝ 〈𝑦, 0R〉)) ↔ (𝐴 <ℝ 〈𝑦, 0R〉 → (𝐴 <ℝ 〈𝑧, 0R〉 ∨ 〈𝑧, 0R〉 <ℝ 〈𝑦, 0R〉)))) |
8 | breq2 3855 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 𝐵)) | |
9 | breq2 3855 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (〈𝑧, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 〈𝑧, 0R〉 <ℝ 𝐵)) | |
10 | 9 | orbi2d 740 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((𝐴 <ℝ 〈𝑧, 0R〉 ∨ 〈𝑧, 0R〉 <ℝ 〈𝑦, 0R〉) ↔ (𝐴 <ℝ 〈𝑧, 0R〉 ∨ 〈𝑧, 0R〉 <ℝ 𝐵))) |
11 | 8, 10 | imbi12d 233 | . 2 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((𝐴 <ℝ 〈𝑦, 0R〉 → (𝐴 <ℝ 〈𝑧, 0R〉 ∨ 〈𝑧, 0R〉 <ℝ 〈𝑦, 0R〉)) ↔ (𝐴 <ℝ 𝐵 → (𝐴 <ℝ 〈𝑧, 0R〉 ∨ 〈𝑧, 0R〉 <ℝ 𝐵)))) |
12 | breq2 3855 | . . . 4 ⊢ (〈𝑧, 0R〉 = 𝐶 → (𝐴 <ℝ 〈𝑧, 0R〉 ↔ 𝐴 <ℝ 𝐶)) | |
13 | breq1 3854 | . . . 4 ⊢ (〈𝑧, 0R〉 = 𝐶 → (〈𝑧, 0R〉 <ℝ 𝐵 ↔ 𝐶 <ℝ 𝐵)) | |
14 | 12, 13 | orbi12d 743 | . . 3 ⊢ (〈𝑧, 0R〉 = 𝐶 → ((𝐴 <ℝ 〈𝑧, 0R〉 ∨ 〈𝑧, 0R〉 <ℝ 𝐵) ↔ (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵))) |
15 | 14 | imbi2d 229 | . 2 ⊢ (〈𝑧, 0R〉 = 𝐶 → ((𝐴 <ℝ 𝐵 → (𝐴 <ℝ 〈𝑧, 0R〉 ∨ 〈𝑧, 0R〉 <ℝ 𝐵)) ↔ (𝐴 <ℝ 𝐵 → (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵)))) |
16 | ltsosr 7371 | . . . 4 ⊢ <R Or R | |
17 | sowlin 4156 | . . . 4 ⊢ (( <R Or R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R)) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦))) | |
18 | 16, 17 | mpan 416 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦))) |
19 | ltresr 7437 | . . 3 ⊢ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝑥 <R 𝑦) | |
20 | ltresr 7437 | . . . 4 ⊢ (〈𝑥, 0R〉 <ℝ 〈𝑧, 0R〉 ↔ 𝑥 <R 𝑧) | |
21 | ltresr 7437 | . . . 4 ⊢ (〈𝑧, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝑧 <R 𝑦) | |
22 | 20, 21 | orbi12i 717 | . . 3 ⊢ ((〈𝑥, 0R〉 <ℝ 〈𝑧, 0R〉 ∨ 〈𝑧, 0R〉 <ℝ 〈𝑦, 0R〉) ↔ (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦)) |
23 | 18, 19, 22 | 3imtr4g 204 | . 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 → (〈𝑥, 0R〉 <ℝ 〈𝑧, 0R〉 ∨ 〈𝑧, 0R〉 <ℝ 〈𝑦, 0R〉))) |
24 | 1, 2, 3, 7, 11, 15, 23 | 3gencl 2654 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 <ℝ 𝐵 → (𝐴 <ℝ 𝐶 ∨ 𝐶 <ℝ 𝐵))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∨ wo 665 ∧ w3a 925 = wceq 1290 ∈ wcel 1439 〈cop 3453 class class class wbr 3851 Or wor 4131 Rcnr 6917 0Rc0r 6918 <R cltr 6923 ℝcr 7410 <ℝ cltrr 7415 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-coll 3960 ax-sep 3963 ax-nul 3971 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-iinf 4416 |
This theorem depends on definitions: df-bi 116 df-dc 782 df-3or 926 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-ral 2365 df-rex 2366 df-reu 2367 df-rab 2369 df-v 2622 df-sbc 2842 df-csb 2935 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-nul 3288 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-int 3695 df-iun 3738 df-br 3852 df-opab 3906 df-mpt 3907 df-tr 3943 df-eprel 4125 df-id 4129 df-po 4132 df-iso 4133 df-iord 4202 df-on 4204 df-suc 4207 df-iom 4419 df-xp 4458 df-rel 4459 df-cnv 4460 df-co 4461 df-dm 4462 df-rn 4463 df-res 4464 df-ima 4465 df-iota 4993 df-fun 5030 df-fn 5031 df-f 5032 df-f1 5033 df-fo 5034 df-f1o 5035 df-fv 5036 df-ov 5669 df-oprab 5670 df-mpt2 5671 df-1st 5925 df-2nd 5926 df-recs 6084 df-irdg 6149 df-1o 6195 df-2o 6196 df-oadd 6199 df-omul 6200 df-er 6306 df-ec 6308 df-qs 6312 df-ni 6924 df-pli 6925 df-mi 6926 df-lti 6927 df-plpq 6964 df-mpq 6965 df-enq 6967 df-nqqs 6968 df-plqqs 6969 df-mqqs 6970 df-1nqqs 6971 df-rq 6972 df-ltnqqs 6973 df-enq0 7044 df-nq0 7045 df-0nq0 7046 df-plq0 7047 df-mq0 7048 df-inp 7086 df-i1p 7087 df-iplp 7088 df-iltp 7090 df-enr 7333 df-nr 7334 df-ltr 7337 df-0r 7338 df-r 7421 df-lt 7424 |
This theorem is referenced by: (None) |
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