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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 8035 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) | |
4 | 1, 2, 3 | syl2anc 411 | 1 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 105 ∈ wcel 2148 class class class wbr 4005 ℝcr 7812 < clt 7994 ≤ cle 7995 |
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-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 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-ral 2460 df-rex 2461 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-br 4006 df-opab 4067 df-xp 4634 df-cnv 4636 df-xr 7998 df-le 8000 |
This theorem is referenced by: ltnsymd 8079 nltled 8080 lensymd 8081 leadd1 8389 lemul1 8552 leltap 8584 ap0gt0 8599 prodgt0 8811 prodge0 8813 lediv1 8828 lemuldiv 8840 lerec 8843 lt2msq 8845 le2msq 8860 squeeze0 8863 suprleubex 8913 0mnnnnn0 9210 elnn0z 9268 uzm1 9560 infregelbex 9600 fztri3or 10041 fzdisj 10054 uzdisj 10095 nn0disj 10140 fzouzdisj 10182 elfzonelfzo 10232 flqeqceilz 10320 modifeq2int 10388 modsumfzodifsn 10398 nn0leexp2 10692 expcanlem 10697 fimaxq 10809 resqrexlemoverl 11032 leabs 11085 absle 11100 maxleast 11224 minmax 11240 climge0 11335 pcfac 12350 cxple 14422 |
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