Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 11046. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 10945. (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 10887 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 10887 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | breq1 5077 | . . 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 5078 | . . . . 5 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 〈𝑦, 0R〉 <ℝ 𝐴)) | |
6 | 4, 5 | orbi12d 916 | . . . 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 346 | . 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 5078 | . . 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 5077 | . . . . 5 ⊢ (〈𝑦, 0R〉 = 𝐵 → (〈𝑦, 0R〉 <ℝ 𝐴 ↔ 𝐵 <ℝ 𝐴)) | |
12 | 10, 11 | orbi12d 916 | . . . 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 346 | . 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 10850 | . . . 4 ⊢ <R Or R | |
16 | sotric 5531 | . . . 4 ⊢ (( <R Or R ∧ (𝑥 ∈ R ∧ 𝑦 ∈ R)) → (𝑥 <R 𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 <R 𝑥))) | |
17 | 15, 16 | mpan 687 | . . 3 ⊢ ((𝑥 ∈ R ∧ 𝑦 ∈ R) → (𝑥 <R 𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦 <R 𝑥))) |
18 | ltresr 10896 | . . 3 ⊢ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝑥 <R 𝑦) | |
19 | vex 3436 | . . . . . 6 ⊢ 𝑥 ∈ V | |
20 | 19 | eqresr 10893 | . . . . 5 ⊢ (〈𝑥, 0R〉 = 〈𝑦, 0R〉 ↔ 𝑥 = 𝑦) |
21 | ltresr 10896 | . . . . 5 ⊢ (〈𝑦, 0R〉 <ℝ 〈𝑥, 0R〉 ↔ 𝑦 <R 𝑥) | |
22 | 20, 21 | orbi12i 912 | . . . 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 3472 | 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 205 ∧ wa 396 ∨ wo 844 = wceq 1539 ∈ wcel 2106 〈cop 4567 class class class wbr 5074 Or wor 5502 Rcnr 10621 0Rc0r 10622 <R cltr 10627 ℝcr 10870 <ℝ cltrr 10875 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-oadd 8301 df-omul 8302 df-er 8498 df-ec 8500 df-qs 8504 df-ni 10628 df-pli 10629 df-mi 10630 df-lti 10631 df-plpq 10664 df-mpq 10665 df-ltpq 10666 df-enq 10667 df-nq 10668 df-erq 10669 df-plq 10670 df-mq 10671 df-1nq 10672 df-rq 10673 df-ltnq 10674 df-np 10737 df-1p 10738 df-plp 10739 df-ltp 10741 df-enr 10811 df-nr 10812 df-ltr 10815 df-0r 10816 df-r 10881 df-lt 10884 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |