Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > ltled | GIF version |
Description: 'Less than' implies 'less than or equal to'. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
ltled.1 | ⊢ (𝜑 → 𝐴 < 𝐵) |
Ref | Expression |
---|---|
ltled | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltled.1 | . 2 ⊢ (𝜑 → 𝐴 < 𝐵) | |
2 | ltd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | ltd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | ltle 7851 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) | |
5 | 2, 3, 4 | syl2anc 408 | . 2 ⊢ (𝜑 → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) |
6 | 1, 5 | mpd 13 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 class class class wbr 3929 ℝcr 7619 < clt 7800 ≤ cle 7801 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-pre-ltirr 7732 ax-pre-lttrn 7734 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-xp 4545 df-cnv 4547 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 |
This theorem is referenced by: ltnsymd 7882 addgt0d 8283 lt2addd 8329 lt2msq1 8643 lediv12a 8652 ledivp1 8661 nn2ge 8753 fznatpl1 9856 exbtwnzlemex 10027 apbtwnz 10047 iseqf1olemkle 10257 expnbnd 10415 cvg1nlemres 10757 resqrexlemnm 10790 resqrexlemcvg 10791 resqrexlemglsq 10794 sqrtgt0 10806 leabs 10846 ltabs 10859 abslt 10860 absle 10861 maxabslemab 10978 2zsupmax 10997 xrmaxiflemab 11016 fsum3cvg3 11165 divcnv 11266 expcnvre 11272 absltap 11278 cvgratnnlemnexp 11293 cvgratnnlemmn 11294 cvgratnnlemfm 11298 mertenslemi1 11304 cos12dec 11474 dvdslelemd 11541 divalglemnn 11615 divalglemeuneg 11620 lcmgcdlem 11758 znege1 11856 sqrt2irraplemnn 11857 ennnfonelemex 11927 strleund 12047 suplociccreex 12771 ivthinclemlm 12781 ivthinclemum 12782 ivthinclemlopn 12783 ivthinclemuopn 12785 ivthdec 12791 dveflem 12855 sin0pilem1 12862 sin0pilem2 12863 coseq0negpitopi 12917 tangtx 12919 cosq34lt1 12931 cos02pilt1 12932 |
Copyright terms: Public domain | W3C validator |