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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 8804 | . 2 ⊢ (𝐴 ∈ ℝ → 𝐴 < (𝐴 + 1)) | |
2 | peano2re 8096 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 + 1) ∈ ℝ) | |
3 | ltle 8048 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝐴 + 1) ∈ ℝ) → (𝐴 < (𝐴 + 1) → 𝐴 ≤ (𝐴 + 1))) | |
4 | 2, 3 | mpdan 421 | . 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 2148 class class class wbr 4005 (class class class)co 5878 ℝcr 7813 1c1 7815 + caddc 7817 < clt 7995 ≤ cle 7996 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 ax-un 4435 ax-setind 4538 ax-cnex 7905 ax-resscn 7906 ax-1cn 7907 ax-1re 7908 ax-icn 7909 ax-addcl 7910 ax-addrcl 7911 ax-mulcl 7912 ax-addcom 7914 ax-addass 7916 ax-i2m1 7919 ax-0lt1 7920 ax-0id 7922 ax-rnegex 7923 ax-pre-ltirr 7926 ax-pre-lttrn 7928 ax-pre-ltadd 7930 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-xp 4634 df-cnv 4636 df-iota 5180 df-fv 5226 df-ov 5881 df-pnf 7997 df-mnf 7998 df-xr 7999 df-ltxr 8000 df-le 8001 |
This theorem is referenced by: p1le 8809 lep1d 8891 peano2uz2 9363 |
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