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Mirrors > Home > ILE Home > Th. List > ltnsymd | 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 |
---|---|
ltnsymd | ⊢ (𝜑 → ¬ 𝐵 < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | ltd.2 | . . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ) | |
3 | ltled.1 | . . 3 ⊢ (𝜑 → 𝐴 < 𝐵) | |
4 | 1, 2, 3 | ltled 8072 | . 2 ⊢ (𝜑 → 𝐴 ≤ 𝐵) |
5 | 1, 2 | lenltd 8071 | . 2 ⊢ (𝜑 → (𝐴 ≤ 𝐵 ↔ ¬ 𝐵 < 𝐴)) |
6 | 4, 5 | mpbid 147 | 1 ⊢ (𝜑 → ¬ 𝐵 < 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∈ wcel 2148 class class class wbr 4002 ℝcr 7807 < clt 7988 ≤ cle 7989 |
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-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7899 ax-resscn 7900 ax-pre-ltirr 7920 ax-pre-lttrn 7922 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 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-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4003 df-opab 4064 df-xp 4631 df-cnv 4633 df-pnf 7990 df-mnf 7991 df-xr 7992 df-ltxr 7993 df-le 7994 |
This theorem is referenced by: frec2uzlt2d 10399 resqrexlemgt0 11022 resqrexlemoverl 11023 cvgratz 11533 ivthinclemuopn 13987 ivthinclemdisj 13989 cnplimclemle 14008 efltlemlt 14066 |
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