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Mirrors > Home > MPE Home > Th. List > axpre-lttri | Structured version Visualization version GIF version |
Description: Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttri 10783. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 10682. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axpre-lttri | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elreal 10624 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 10624 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | breq1 5030 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 〈𝑦, 0R〉)) | |
4 | eqeq1 2742 | . . . . 5 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ 𝐴 = 〈𝑦, 0R〉)) | |
5 | breq2 5031 | . . . . 5 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 〈𝑦, 0R〉 <ℝ 𝐴)) | |
6 | 4, 5 | orbi12d 918 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((〈𝑥, 0R〉 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ (𝐴 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴))) |
7 | 6 | notbid 321 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (¬ (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ ¬ (𝐴 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴))) |
8 | 3, 7 | bibi12d 349 | . 2 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ ¬ (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉)) ↔ (𝐴 <ℝ 〈𝑦, 0R〉 ↔ ¬ (𝐴 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴)))) |
9 | breq2 5031 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 𝐵)) | |
10 | eqeq2 2750 | . . . . 5 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 = 〈𝑦, 0R〉 ↔ 𝐴 = 𝐵)) | |
11 | breq1 5030 | . . . . 5 ⊢ (〈𝑦, 0R〉 = 𝐵 → (〈𝑦, 0R〉 <ℝ 𝐴 ↔ 𝐵 <ℝ 𝐴)) | |
12 | 10, 11 | orbi12d 918 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((𝐴 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴) ↔ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴))) |
13 | 12 | notbid 321 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → (¬ (𝐴 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴))) |
14 | 9, 13 | bibi12d 349 | . 2 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((𝐴 <ℝ 〈𝑦, 0R〉 ↔ ¬ (𝐴 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴)) ↔ (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴)))) |
15 | ltsosr 10587 | . . . 4 ⊢ <R Or R | |
16 | sotric 5465 | . . . 4 ⊢ (( <R Or R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) → (𝑥 <R 𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 <R 𝑥))) | |
17 | 15, 16 | mpan 690 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 <R 𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 <R 𝑥))) |
18 | ltresr 10633 | . . 3 ⊢ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝑥 <R 𝑦) | |
19 | vex 3401 | . . . . . 6 ⊢ 𝑥 ∈ V | |
20 | 19 | eqresr 10630 | . . . . 5 ⊢ (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ 𝑥 = 𝑦) |
21 | ltresr 10633 | . . . . 5 ⊢ (〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 𝑦 <R 𝑥) | |
22 | 20, 21 | orbi12i 914 | . . . 4 ⊢ ((〈𝑥, 0R〉 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ (𝑥 = 𝑦 ∨ 𝑦 <R 𝑥)) |
23 | 22 | notbii 323 | . . 3 ⊢ (¬ (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 <R 𝑥)) |
24 | 17, 18, 23 | 3bitr4g 317 | . 2 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ ¬ (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉))) |
25 | 1, 2, 8, 14, 24 | 2gencl 3437 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 846 = wceq 1542 ∈ wcel 2113 〈cop 4519 class class class wbr 5027 Or wor 5437 Rcnr 10358 0Rc0r 10359 <R cltr 10364 ℝcr 10607 <ℝ cltrr 10612 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 ax-inf2 9170 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6123 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-wrecs 7969 df-recs 8030 df-rdg 8068 df-1o 8124 df-oadd 8128 df-omul 8129 df-er 8313 df-ec 8315 df-qs 8319 df-ni 10365 df-pli 10366 df-mi 10367 df-lti 10368 df-plpq 10401 df-mpq 10402 df-ltpq 10403 df-enq 10404 df-nq 10405 df-erq 10406 df-plq 10407 df-mq 10408 df-1nq 10409 df-rq 10410 df-ltnq 10411 df-np 10474 df-1p 10475 df-plp 10476 df-ltp 10478 df-enr 10548 df-nr 10549 df-ltr 10552 df-0r 10553 df-r 10618 df-lt 10621 |
This theorem is referenced by: (None) |
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