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Mirrors > Home > ILE Home > Th. List > lelttrdi | GIF version |
Description: If a number is less than another number, and the other number is less than or equal to a third number, the first number is less than the third number. (Contributed by Alexander van der Vekens, 24-Mar-2018.) |
Ref | Expression |
---|---|
lelttrdi.r | ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) |
lelttrdi.l | ⊢ (𝜑 → 𝐵 ≤ 𝐶) |
Ref | Expression |
---|---|
lelttrdi | ⊢ (𝜑 → (𝐴 < 𝐵 → 𝐴 < 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lelttrdi.r | . . . . 5 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ)) | |
2 | 1 | simp1d 993 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
3 | 2 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐴 ∈ ℝ) |
4 | 1 | simp2d 994 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℝ) |
5 | 4 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐵 ∈ ℝ) |
6 | 1 | simp3d 995 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
7 | 6 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐶 ∈ ℝ) |
8 | simpr 109 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐴 < 𝐵) | |
9 | lelttrdi.l | . . . 4 ⊢ (𝜑 → 𝐵 ≤ 𝐶) | |
10 | 9 | adantr 274 | . . 3 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐵 ≤ 𝐶) |
11 | 3, 5, 7, 8, 10 | ltletrd 8178 | . 2 ⊢ ((𝜑 ∧ 𝐴 < 𝐵) → 𝐴 < 𝐶) |
12 | 11 | ex 114 | 1 ⊢ (𝜑 → (𝐴 < 𝐵 → 𝐴 < 𝐶)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 962 ∈ wcel 1480 class class class wbr 3924 ℝcr 7612 < clt 7793 ≤ cle 7794 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltwlin 7726 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-xp 4540 df-cnv 4542 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 |
This theorem is referenced by: subfzo0 10012 |
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