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Mirrors > Home > ILE Home > Th. List > ltleii | GIF version |
Description: 'Less than' implies 'less than or equal to' (inference). (Contributed by NM, 22-Aug-1999.) |
Ref | Expression |
---|---|
lt.1 | ⊢ 𝐴 ∈ ℝ |
lt.2 | ⊢ 𝐵 ∈ ℝ |
ltlei.1 | ⊢ 𝐴 < 𝐵 |
Ref | Expression |
---|---|
ltleii | ⊢ 𝐴 ≤ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltlei.1 | . 2 ⊢ 𝐴 < 𝐵 | |
2 | lt.1 | . . 3 ⊢ 𝐴 ∈ ℝ | |
3 | lt.2 | . . 3 ⊢ 𝐵 ∈ ℝ | |
4 | 2, 3 | ltlei 7833 | . 2 ⊢ (𝐴 < 𝐵 → 𝐴 ≤ 𝐵) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ 𝐴 ≤ 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 class class class wbr 3899 ℝcr 7587 < clt 7768 ≤ cle 7769 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-pre-ltirr 7700 ax-pre-lttrn 7702 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-xp 4515 df-cnv 4517 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 |
This theorem is referenced by: 0le1 8211 1le2 8896 1le3 8899 halfge0 8904 decleh 9184 uzuzle23 9334 fzo0to42pr 9965 4bc2eq6 10488 resqrexlemga 10763 sqrt9 10788 sqrt2gt1lt2 10789 sqrtpclii 10870 0.999... 11258 ef01bndlem 11390 sin01bnd 11391 cos01bnd 11392 cos2bnd 11394 cos12dec 11401 flodddiv4 11558 strleun 11975 dveflem 12782 sinhalfpilem 12799 sincosq1lem 12833 sincos4thpi 12848 sincos6thpi 12850 pigt3 12852 pige3 12853 cosq34lt1 12858 cos02pilt1 12859 ex-fl 12864 ex-gcd 12870 |
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