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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 7889 | . 2 ⊢ (𝐴 < 𝐵 → 𝐴 ≤ 𝐵) |
5 | 1, 4 | ax-mp 5 | 1 ⊢ 𝐴 ≤ 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1481 class class class wbr 3937 ℝcr 7643 < clt 7824 ≤ cle 7825 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-pre-ltirr 7756 ax-pre-lttrn 7758 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-xp 4553 df-cnv 4555 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 |
This theorem is referenced by: 0le1 8267 1le2 8952 1le3 8955 halfge0 8960 decleh 9240 eluz4eluz2 9389 uzuzle23 9393 fzo0to42pr 10028 4bc2eq6 10552 resqrexlemga 10827 sqrt9 10852 sqrt2gt1lt2 10853 sqrtpclii 10934 0.999... 11322 ef01bndlem 11499 sin01bnd 11500 cos01bnd 11501 cos2bnd 11503 cos12dec 11510 flodddiv4 11667 strleun 12087 dveflem 12895 sinhalfpilem 12920 sincosq1lem 12954 sincos4thpi 12969 sincos6thpi 12971 pigt3 12973 pige3 12974 cosq34lt1 12979 cos02pilt1 12980 cos0pilt1 12981 rpabscxpbnd 13067 2logb9irr 13096 2logb9irrap 13102 ex-fl 13108 ex-gcd 13114 |
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