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Mirrors > Home > ILE Home > Th. List > lep1 | GIF version |
Description: A number is less than or equal to itself plus 1. (Contributed by NM, 5-Jan-2006.) |
Ref | Expression |
---|---|
lep1 | ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ (𝐴 + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltp1 8359 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1)) | |
2 | peano2re 7672 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ) | |
3 | ltle 7626 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐴 + 1) ∈ ℝ) → (𝐴 < (𝐴 + 1) → 𝐴 ≤ (𝐴 + 1))) | |
4 | 2, 3 | mpdan 413 | . 2 ⊢ (𝐴 ∈ ℝ → (𝐴 < (𝐴 + 1) → 𝐴 ≤ (𝐴 + 1))) |
5 | 1, 4 | mpd 13 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 ≤ (𝐴 + 1)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1439 class class class wbr 3851 (class class class)co 5666 ℝcr 7403 1c1 7405 + caddc 7407 < clt 7576 ≤ cle 7577 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 580 ax-in2 581 ax-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-13 1450 ax-14 1451 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 ax-sep 3963 ax-pow 4015 ax-pr 4045 ax-un 4269 ax-setind 4366 ax-cnex 7490 ax-resscn 7491 ax-1cn 7492 ax-1re 7493 ax-icn 7494 ax-addcl 7495 ax-addrcl 7496 ax-mulcl 7497 ax-addcom 7499 ax-addass 7501 ax-i2m1 7504 ax-0lt1 7505 ax-0id 7507 ax-rnegex 7508 ax-pre-ltirr 7511 ax-pre-lttrn 7513 ax-pre-ltadd 7515 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-fal 1296 df-nf 1396 df-sb 1694 df-eu 1952 df-mo 1953 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-ne 2257 df-nel 2352 df-ral 2365 df-rex 2366 df-rab 2369 df-v 2622 df-dif 3002 df-un 3004 df-in 3006 df-ss 3013 df-pw 3435 df-sn 3456 df-pr 3457 df-op 3459 df-uni 3660 df-br 3852 df-opab 3906 df-xp 4457 df-cnv 4459 df-iota 4993 df-fv 5036 df-ov 5669 df-pnf 7578 df-mnf 7579 df-xr 7580 df-ltxr 7581 df-le 7582 |
This theorem is referenced by: p1le 8364 lep1d 8446 peano2uz2 8907 |
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