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Mirrors > Home > ILE Home > Th. List > lenltd | GIF version |
Description: 'Less than or equal to' in terms of 'less than'. (Contributed by Mario Carneiro, 27-May-2016.) |
Ref | Expression |
---|---|
ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
Ref | Expression |
---|---|
lenltd | ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltd.2 | . 2 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | lenlt 7965 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
4 | 1, 2, 3 | syl2anc 409 | 1 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 104 ∈ wcel 2135 class class class wbr 3976 ℝcr 7743 < clt 7924 ≤ cle 7925 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-dif 3113 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-xp 4604 df-cnv 4606 df-xr 7928 df-le 7930 |
This theorem is referenced by: ltnsymd 8009 nltled 8010 lensymd 8011 leadd1 8319 lemul1 8482 leltap 8514 ap0gt0 8529 prodgt0 8738 prodge0 8740 lediv1 8755 lemuldiv 8767 lerec 8770 lt2msq 8772 le2msq 8787 squeeze0 8790 suprleubex 8840 0mnnnnn0 9137 elnn0z 9195 uzm1 9487 infregelbex 9527 fztri3or 9964 fzdisj 9977 uzdisj 10018 nn0disj 10063 fzouzdisj 10105 elfzonelfzo 10155 flqeqceilz 10243 modifeq2int 10311 modsumfzodifsn 10321 nn0leexp2 10613 expcanlem 10617 fimaxq 10729 resqrexlemoverl 10949 leabs 11002 absle 11017 maxleast 11141 minmax 11157 climge0 11252 pcfac 12259 cxple 13384 |
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