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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 11332. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 11229. (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 11171 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
| 2 | elreal 11171 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
| 3 | breq1 5146 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 〈𝑦, 0R〉)) | |
| 4 | eqeq1 2741 | . . . . 5 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ 𝐴 = 〈𝑦, 0R〉)) | |
| 5 | breq2 5147 | . . . . 5 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 〈𝑦, 0R〉 <ℝ 𝐴)) | |
| 6 | 4, 5 | orbi12d 919 | . . . 4 ⊢ (〈𝑥, 0R〉 = 𝐴 → ((〈𝑥, 0R〉 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ (𝐴 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴))) |
| 7 | 6 | notbid 318 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (¬ (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ ¬ (𝐴 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴))) |
| 8 | 3, 7 | bibi12d 345 | . 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 5147 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 𝐵)) | |
| 10 | eqeq2 2749 | . . . . 5 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 = 〈𝑦, 0R〉 ↔ 𝐴 = 𝐵)) | |
| 11 | breq1 5146 | . . . . 5 ⊢ (〈𝑦, 0R〉 = 𝐵 → (〈𝑦, 0R〉 <ℝ 𝐴 ↔ 𝐵 <ℝ 𝐴)) | |
| 12 | 10, 11 | orbi12d 919 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → ((𝐴 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴) ↔ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴))) |
| 13 | 12 | notbid 318 | . . 3 ⊢ (〈𝑦, 0R〉 = 𝐵 → (¬ (𝐴 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 𝐴) ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 <ℝ 𝐴))) |
| 14 | 9, 13 | bibi12d 345 | . 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 11134 | . . . 4 ⊢ <R Or R | |
| 16 | sotric 5622 | . . . 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 11180 | . . 3 ⊢ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝑥 <R 𝑦) | |
| 19 | vex 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 20 | 19 | eqresr 11177 | . . . . 5 ⊢ (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ 𝑥 = 𝑦) |
| 21 | ltresr 11180 | . . . . 5 ⊢ (〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 𝑦 <R 𝑥) | |
| 22 | 20, 21 | orbi12i 915 | . . . 4 ⊢ ((〈𝑥, 0R〉 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ (𝑥 = 𝑦 ∨ 𝑦 <R 𝑥)) |
| 23 | 22 | notbii 320 | . . 3 ⊢ (¬ (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ∨ 〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉) ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 <R 𝑥)) |
| 24 | 17, 18, 23 | 3bitr4g 314 | . 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 3524 | 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 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 〈cop 4632 class class class wbr 5143 Or wor 5591 Rcnr 10905 0Rc0r 10906 <R cltr 10911 ℝcr 11154 <ℝ cltrr 11159 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-oadd 8510 df-omul 8511 df-er 8745 df-ec 8747 df-qs 8751 df-ni 10912 df-pli 10913 df-mi 10914 df-lti 10915 df-plpq 10948 df-mpq 10949 df-ltpq 10950 df-enq 10951 df-nq 10952 df-erq 10953 df-plq 10954 df-mq 10955 df-1nq 10956 df-rq 10957 df-ltnq 10958 df-np 11021 df-1p 11022 df-plp 11023 df-ltp 11025 df-enr 11095 df-nr 11096 df-ltr 11099 df-0r 11100 df-r 11165 df-lt 11168 |
| This theorem is referenced by: (None) |
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