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Mirrors > Home > ILE Home > Th. List > p1le | GIF version |
Description: A transitive property of plus 1 and 'less than or equal'. (Contributed by NM, 16-Aug-2005.) |
Ref | Expression |
---|---|
p1le | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + 1) ≤ 𝐵) → 𝐴 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lep1 8060 | . . . 4 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ (𝐴 + 1)) | |
2 | 1 | adantr 270 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ≤ (𝐴 + 1)) |
3 | peano2re 7381 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ) | |
4 | 3 | ancli 316 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 ∈ ℝ ∧ (𝐴 + 1) ∈ ℝ)) |
5 | letr 7331 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝐴 + 1) ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 ≤ (𝐴 + 1) ∧ (𝐴 + 1) ≤ 𝐵) → 𝐴 ≤ 𝐵)) | |
6 | 5 | 3expa 1139 | . . . 4 ⊢ (((𝐴 ∈ ℝ ∧ (𝐴 + 1) ∈ ℝ) ∧ 𝐵 ∈ ℝ) → ((𝐴 ≤ (𝐴 + 1) ∧ (𝐴 + 1) ≤ 𝐵) → 𝐴 ≤ 𝐵)) |
7 | 4, 6 | sylan 277 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 ≤ (𝐴 + 1) ∧ (𝐴 + 1) ≤ 𝐵) → 𝐴 ≤ 𝐵)) |
8 | 2, 7 | mpand 420 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ((𝐴 + 1) ≤ 𝐵 → 𝐴 ≤ 𝐵)) |
9 | 8 | 3impia 1136 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝐴 + 1) ≤ 𝐵) → 𝐴 ≤ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 920 ∈ wcel 1434 class class class wbr 3805 (class class class)co 5564 ℝcr 7112 1c1 7114 + caddc 7116 ≤ cle 7286 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3916 ax-pow 3968 ax-pr 3992 ax-un 4216 ax-setind 4308 ax-cnex 7199 ax-resscn 7200 ax-1cn 7201 ax-1re 7202 ax-icn 7203 ax-addcl 7204 ax-addrcl 7205 ax-mulcl 7206 ax-addcom 7208 ax-addass 7210 ax-i2m1 7213 ax-0lt1 7214 ax-0id 7216 ax-rnegex 7217 ax-pre-ltirr 7220 ax-pre-ltwlin 7221 ax-pre-lttrn 7222 ax-pre-ltadd 7224 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2612 df-dif 2984 df-un 2986 df-in 2988 df-ss 2995 df-pw 3402 df-sn 3422 df-pr 3423 df-op 3425 df-uni 3622 df-br 3806 df-opab 3860 df-xp 4397 df-cnv 4399 df-iota 4917 df-fv 4960 df-ov 5567 df-pnf 7287 df-mnf 7288 df-xr 7289 df-ltxr 7290 df-le 7291 |
This theorem is referenced by: fzind 8613 |
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