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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 7845. (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 7748 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 7748 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | elreal 7748 | . 2 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R 〈𝑧, 0R〉 = 𝐶) | |
4 | breq1 3968 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 〈𝑦, 0R〉)) | |
5 | breq1 3968 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 <ℝ 〈𝑧, 0R〉 ↔ 𝐴 <ℝ 〈𝑧, 0R〉)) | |
6 | 5 | orbi1d 781 | . . 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 3969 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 𝐵)) | |
9 | breq2 3969 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (〈𝑧, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 〈𝑧, 0R〉 <ℝ 𝐵)) | |
10 | 9 | orbi2d 780 | . . 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 3969 | . . . 4 ⊢ (〈𝑧, 0R〉 = 𝐶 → (𝐴 <ℝ 〈𝑧, 0R〉 ↔ 𝐴 <ℝ 𝐶)) | |
13 | breq1 3968 | . . . 4 ⊢ (〈𝑧, 0R〉 = 𝐶 → (〈𝑧, 0R〉 <ℝ 𝐵 ↔ 𝐶 <ℝ 𝐵)) | |
14 | 12, 13 | orbi12d 783 | . . 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 7684 | . . . 4 ⊢ <R Or R | |
17 | sowlin 4280 | . . . 4 ⊢ (( <R Or R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R)) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦))) | |
18 | 16, 17 | mpan 421 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → (𝑥 <R 𝑦 → (𝑥 <R 𝑧 ∨ 𝑧 <R 𝑦))) |
19 | ltresr 7759 | . . 3 ⊢ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝑥 <R 𝑦) | |
20 | ltresr 7759 | . . . 4 ⊢ (〈𝑥, 0R〉 <ℝ 〈𝑧, 0R〉 ↔ 𝑥 <R 𝑧) | |
21 | ltresr 7759 | . . . 4 ⊢ (〈𝑧, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝑧 <R 𝑦) | |
22 | 20, 21 | orbi12i 754 | . . 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 2746 | 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 698 ∧ w3a 963 = wceq 1335 ∈ wcel 2128 〈cop 3563 class class class wbr 3965 Or wor 4255 Rcnr 7217 0Rc0r 7218 <R cltr 7223 ℝcr 7731 <ℝ cltrr 7736 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-eprel 4249 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-recs 6252 df-irdg 6317 df-1o 6363 df-2o 6364 df-oadd 6367 df-omul 6368 df-er 6480 df-ec 6482 df-qs 6486 df-ni 7224 df-pli 7225 df-mi 7226 df-lti 7227 df-plpq 7264 df-mpq 7265 df-enq 7267 df-nqqs 7268 df-plqqs 7269 df-mqqs 7270 df-1nqqs 7271 df-rq 7272 df-ltnqqs 7273 df-enq0 7344 df-nq0 7345 df-0nq0 7346 df-plq0 7347 df-mq0 7348 df-inp 7386 df-i1p 7387 df-iplp 7388 df-iltp 7390 df-enr 7646 df-nr 7647 df-ltr 7650 df-0r 7651 df-r 7742 df-lt 7745 |
This theorem is referenced by: (None) |
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