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Mirrors > Home > MPE Home > Th. List > axpre-lttrn | Structured version Visualization version GIF version |
Description: Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttrn 10712. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 10611. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axpre-lttrn | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elreal 10552 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 10552 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | elreal 10552 | . 2 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R 〈𝑧, 0R〉 = 𝐶) | |
4 | breq1 5068 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 〈𝑦, 0R〉)) | |
5 | 4 | anbi1d 631 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∧ 〈𝑦, 0R〉 <ℝ 〈𝑧, 0R〉) ↔ (𝐴 <ℝ 〈𝑦, 0R〉 ∧ 〈𝑦, 0R〉 <ℝ 〈𝑧, 0R〉))) |
6 | breq1 5068 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 <ℝ 〈𝑧, 0R〉 ↔ 𝐴 <ℝ 〈𝑧, 0R〉)) | |
7 | 5, 6 | imbi12d 347 | . 2 ⊢ (〈𝑥, 0R〉 = 𝐴 → (((〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∧ 〈𝑦, 0R〉 <ℝ 〈𝑧, 0R〉) → 〈𝑥, 0R〉 <ℝ 〈𝑧, 0R〉) ↔ ((𝐴 <ℝ 〈𝑦, 0R〉 ∧ 〈𝑦, 0R〉 <ℝ 〈𝑧, 0R〉) → 𝐴 <ℝ 〈𝑧, 0R〉))) |
8 | breq2 5069 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 𝐵)) | |
9 | breq1 5068 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (〈𝑦, 0R〉 <ℝ 〈𝑧, 0R〉 ↔ 𝐵 <ℝ 〈𝑧, 0R〉)) | |
10 | 8, 9 | anbi12d 632 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((𝐴 <ℝ 〈𝑦, 0R〉 ∧ 〈𝑦, 0R〉 <ℝ 〈𝑧, 0R〉) ↔ (𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 〈𝑧, 0R〉))) |
11 | 10 | imbi1d 344 | . 2 ⊢ (〈𝑦, 0R〉 = 𝐵 → (((𝐴 <ℝ 〈𝑦, 0R〉 ∧ 〈𝑦, 0R〉 <ℝ 〈𝑧, 0R〉) → 𝐴 <ℝ 〈𝑧, 0R〉) ↔ ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 〈𝑧, 0R〉) → 𝐴 <ℝ 〈𝑧, 0R〉))) |
12 | breq2 5069 | . . . 4 ⊢ (〈𝑧, 0R〉 = 𝐶 → (𝐵 <ℝ 〈𝑧, 0R〉 ↔ 𝐵 <ℝ 𝐶)) | |
13 | 12 | anbi2d 630 | . . 3 ⊢ (〈𝑧, 0R〉 = 𝐶 → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 〈𝑧, 0R〉) ↔ (𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶))) |
14 | breq2 5069 | . . 3 ⊢ (〈𝑧, 0R〉 = 𝐶 → (𝐴 <ℝ 〈𝑧, 0R〉 ↔ 𝐴 <ℝ 𝐶)) | |
15 | 13, 14 | imbi12d 347 | . 2 ⊢ (〈𝑧, 0R〉 = 𝐶 → (((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 〈𝑧, 0R〉) → 𝐴 <ℝ 〈𝑧, 0R〉) ↔ ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶))) |
16 | ltresr 10561 | . . . . 5 ⊢ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝑥 <R 𝑦) | |
17 | ltresr 10561 | . . . . 5 ⊢ (〈𝑦, 0R〉 <ℝ 〈𝑧, 0R〉 ↔ 𝑦 <R 𝑧) | |
18 | ltsosr 10515 | . . . . . 6 ⊢ <R Or R | |
19 | ltrelsr 10489 | . . . . . 6 ⊢ <R ⊆ (R × R) | |
20 | 18, 19 | sotri 5986 | . . . . 5 ⊢ ((𝑥 <R 𝑦 ∧ 𝑦 <R 𝑧) → 𝑥 <R 𝑧) |
21 | 16, 17, 20 | syl2anb 599 | . . . 4 ⊢ ((〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∧ 〈𝑦, 0R〉 <ℝ 〈𝑧, 0R〉) → 𝑥 <R 𝑧) |
22 | ltresr 10561 | . . . 4 ⊢ (〈𝑥, 0R〉 <ℝ 〈𝑧, 0R〉 ↔ 𝑥 <R 𝑧) | |
23 | 21, 22 | sylibr 236 | . . 3 ⊢ ((〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∧ 〈𝑦, 0R〉 <ℝ 〈𝑧, 0R〉) → 〈𝑥, 0R〉 <ℝ 〈𝑧, 0R〉) |
24 | 23 | a1i 11 | . 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R ∧ 𝑧 ∈ R) → ((〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∧ 〈𝑦, 0R〉 <ℝ 〈𝑧, 0R〉) → 〈𝑥, 0R〉 <ℝ 〈𝑧, 0R〉)) |
25 | 1, 2, 3, 7, 11, 15, 24 | 3gencl 3536 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 <ℝ 𝐵 ∧ 𝐵 <ℝ 𝐶) → 𝐴 <ℝ 𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 〈cop 4572 class class class wbr 5065 Rcnr 10286 0Rc0r 10287 <R cltr 10292 ℝcr 10535 <ℝ cltrr 10540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-1st 7688 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-oadd 8105 df-omul 8106 df-er 8288 df-ec 8290 df-qs 8294 df-ni 10293 df-pli 10294 df-mi 10295 df-lti 10296 df-plpq 10329 df-mpq 10330 df-ltpq 10331 df-enq 10332 df-nq 10333 df-erq 10334 df-plq 10335 df-mq 10336 df-1nq 10337 df-rq 10338 df-ltnq 10339 df-np 10402 df-1p 10403 df-plp 10404 df-ltp 10406 df-enr 10476 df-nr 10477 df-ltr 10480 df-0r 10481 df-r 10546 df-lt 10549 |
This theorem is referenced by: (None) |
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