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Mirrors > Home > ILE Home > Th. List > nltled | GIF version |
Description: 'Not less than ' implies 'less than or equal to'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
Ref | Expression |
---|---|
ltd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
ltd.2 | ⊢ (𝜑 → 𝐵 ∈ ℝ) |
nltled.1 | ⊢ (𝜑 → ¬ 𝐵 < 𝐴) |
Ref | Expression |
---|---|
nltled | ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nltled.1 | . 2 ⊢ (𝜑 → ¬ 𝐵 < 𝐴) | |
2 | ltd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
3 | ltd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
4 | 2, 3 | lenltd 8007 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
5 | 1, 4 | mpbird 166 | 1 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ 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: ltntri 8017 suprubex 8837 infregelbex 9527 cvgratz 11459 zsupcl 11865 zssinfcl 11866 infssuzledc 11868 dvdslegcd 11882 pw2dvdseulemle 12076 dedekindeulemuub 13136 dedekindeulemlu 13140 suplociccex 13144 dedekindicclemuub 13145 dedekindicclemlu 13149 ivthinclemlopn 13155 ivthinclemuopn 13157 refeq 13741 |
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