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 7105 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 → 𝐴 ≤ 𝐵)) | |
5 | 2, 3, 4 | syl2anc 391 | . 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 1393 class class class wbr 3764 ℝcr 6888 < clt 7060 ≤ cle 7061 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-pre-ltirr 6996 ax-pre-lttrn 6998 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-nel 2207 df-ral 2311 df-rex 2312 df-rab 2315 df-v 2559 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-xp 4351 df-cnv 4353 df-pnf 7062 df-mnf 7063 df-xr 7064 df-ltxr 7065 df-le 7066 |
This theorem is referenced by: ltnsymd 7136 addgt0d 7512 lt2addd 7558 lt2msq1 7851 lediv12a 7860 ledivp1 7869 nn2ge 7946 fznatpl1 8938 qbtwnzlemex 9105 expnbnd 9372 cvg1nlemres 9584 resqrexlemnm 9616 resqrexlemcvg 9617 resqrexlemglsq 9620 sqrtgt0 9632 leabs 9672 ltabs 9683 abslt 9684 absle 9685 |
Copyright terms: Public domain | W3C validator |