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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 10707. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 10606. (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 10547 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 10547 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | elreal 10547 | . 2 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R 〈𝑧, 0R〉 = 𝐶) | |
4 | breq1 5061 | . . . 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 5061 | . . 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 5062 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 𝐵)) | |
9 | breq1 5061 | . . . 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 5062 | . . . 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 5062 | . . 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 10556 | . . . . 5 ⊢ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝑥 <R 𝑦) | |
17 | ltresr 10556 | . . . . 5 ⊢ (〈𝑦, 0R〉 <ℝ 〈𝑧, 0R〉 ↔ 𝑦 <R 𝑧) | |
18 | ltsosr 10510 | . . . . . 6 ⊢ <R Or R | |
19 | ltrelsr 10484 | . . . . . 6 ⊢ <R ⊆ (R × R) | |
20 | 18, 19 | sotri 5981 | . . . . 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 10556 | . . . 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 4566 class class class wbr 5058 Rcnr 10281 0Rc0r 10282 <R cltr 10287 ℝcr 10530 <ℝ cltrr 10535 |
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 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 |
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 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-omul 8101 df-er 8283 df-ec 8285 df-qs 8289 df-ni 10288 df-pli 10289 df-mi 10290 df-lti 10291 df-plpq 10324 df-mpq 10325 df-ltpq 10326 df-enq 10327 df-nq 10328 df-erq 10329 df-plq 10330 df-mq 10331 df-1nq 10332 df-rq 10333 df-ltnq 10334 df-np 10397 df-1p 10398 df-plp 10399 df-ltp 10401 df-enr 10471 df-nr 10472 df-ltr 10475 df-0r 10476 df-r 10541 df-lt 10544 |
This theorem is referenced by: (None) |
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