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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 6902 | . . 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 1390 class class class wbr 3755 ℝcr 6710 < clt 6857 ≤ cle 6858 |
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 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 ax-pre-ltirr 6795 ax-pre-lttrn 6797 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-rab 2309 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-xp 4294 df-cnv 4296 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-le 6863 |
This theorem is referenced by: ltnsymd 6933 addgt0d 7307 lt2addd 7353 lt2msq1 7632 lediv12a 7641 ledivp1 7650 nn2ge 7727 fznatpl1 8708 expnbnd 9025 |
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