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| Mirrors > Home > MPE Home > Th. List > Mathboxes > lttrii | Structured version Visualization version GIF version | ||
| Description: 'Less than' is transitive. (Contributed by SN, 26-Aug-2025.) |
| Ref | Expression |
|---|---|
| lttrii.a | ⊢ 𝐴 ∈ ℝ |
| lttrii.b | ⊢ 𝐵 ∈ ℝ |
| lttrii.c | ⊢ 𝐶 ∈ ℝ |
| lttrii.1 | ⊢ 𝐴 < 𝐵 |
| lttrii.2 | ⊢ 𝐵 < 𝐶 |
| Ref | Expression |
|---|---|
| lttrii | ⊢ 𝐴 < 𝐶 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | lttrii.1 | . 2 ⊢ 𝐴 < 𝐵 | |
| 2 | lttrii.2 | . 2 ⊢ 𝐵 < 𝐶 | |
| 3 | lttrii.a | . . 3 ⊢ 𝐴 ∈ ℝ | |
| 4 | lttrii.b | . . 3 ⊢ 𝐵 ∈ ℝ | |
| 5 | lttrii.c | . . 3 ⊢ 𝐶 ∈ ℝ | |
| 6 | 3, 4, 5 | lttri 11383 | . 2 ⊢ ((𝐴 < 𝐵 ∧ 𝐵 < 𝐶) → 𝐴 < 𝐶) |
| 7 | 1, 2, 6 | mp2an 692 | 1 ⊢ 𝐴 < 𝐶 |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 class class class wbr 5141 ℝcr 11150 < clt 11291 |
| 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 2707 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 ax-resscn 11208 ax-pre-lttrn 11226 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-er 8741 df-en 8982 df-dom 8983 df-sdom 8984 df-pnf 11293 df-mnf 11294 df-ltxr 11296 |
| This theorem is referenced by: asin1half 42365 |
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