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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 11331. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 11228. (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 11169 | . 2 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑥 ∈ R 〈𝑥, 0R〉 = 𝐴) | |
2 | elreal 11169 | . 2 ⊢ (𝐵 ∈ ℝ ↔ ∃𝑦 ∈ R 〈𝑦, 0R〉 = 𝐵) | |
3 | elreal 11169 | . 2 ⊢ (𝐶 ∈ ℝ ↔ ∃𝑧 ∈ R 〈𝑧, 0R〉 = 𝐶) | |
4 | breq1 5151 | . . . 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 5151 | . . 3 ⊢ (〈𝑥, 0R〉 = 𝐴 → (〈𝑥, 0R〉 <ℝ 〈𝑧, 0R〉 ↔ 𝐴 <ℝ 〈𝑧, 0R〉)) | |
7 | 5, 6 | imbi12d 344 | . 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 5152 | . . . 4 ⊢ (〈𝑦, 0R〉 = 𝐵 → (𝐴 <ℝ 〈𝑦, 0R〉 ↔ 𝐴 <ℝ 𝐵)) | |
9 | breq1 5151 | . . . 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 341 | . 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 5152 | . . . 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 5152 | . . 3 ⊢ (〈𝑧, 0R〉 = 𝐶 → (𝐴 <ℝ 〈𝑧, 0R〉 ↔ 𝐴 <ℝ 𝐶)) | |
15 | 13, 14 | imbi12d 344 | . 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 11178 | . . . . 5 ⊢ (〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ↔ 𝑥 <R 𝑦) | |
17 | ltresr 11178 | . . . . 5 ⊢ (〈𝑦, 0R〉 <ℝ 〈𝑧, 0R〉 ↔ 𝑦 <R 𝑧) | |
18 | ltsosr 11132 | . . . . . 6 ⊢ <R Or R | |
19 | ltrelsr 11106 | . . . . . 6 ⊢ <R ⊆ (R × R) | |
20 | 18, 19 | sotri 6150 | . . . . 5 ⊢ ((𝑥 <R 𝑦 ∧ 𝑦 <R 𝑧) → 𝑥 <R 𝑧) |
21 | 16, 17, 20 | syl2anb 598 | . . . 4 ⊢ ((〈𝑥, 0R〉 <ℝ 〈𝑦, 0R〉 ∧ 〈𝑦, 0R〉 <ℝ 〈𝑧, 0R〉) → 𝑥 <R 𝑧) |
22 | ltresr 11178 | . . . 4 ⊢ (〈𝑥, 0R〉 <ℝ 〈𝑧, 0R〉 ↔ 𝑥 <R 𝑧) | |
23 | 21, 22 | sylibr 234 | . . 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 3523 | 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 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 〈cop 4637 class class class wbr 5148 Rcnr 10903 0Rc0r 10904 <R cltr 10909 ℝcr 11152 <ℝ cltrr 11157 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-oadd 8509 df-omul 8510 df-er 8744 df-ec 8746 df-qs 8750 df-ni 10910 df-pli 10911 df-mi 10912 df-lti 10913 df-plpq 10946 df-mpq 10947 df-ltpq 10948 df-enq 10949 df-nq 10950 df-erq 10951 df-plq 10952 df-mq 10953 df-1nq 10954 df-rq 10955 df-ltnq 10956 df-np 11019 df-1p 11020 df-plp 11021 df-ltp 11023 df-enr 11093 df-nr 11094 df-ltr 11097 df-0r 11098 df-r 11163 df-lt 11166 |
This theorem is referenced by: (None) |
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