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Theorem xrlenltd 11327
Description: "Less than or equal to" expressed in terms of "less than", for extended reals. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
xrlenltd.a (𝜑𝐴 ∈ ℝ*)
xrlenltd.b (𝜑𝐵 ∈ ℝ*)
Assertion
Ref Expression
xrlenltd (𝜑 → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))

Proof of Theorem xrlenltd
StepHypRef Expression
1 xrlenltd.a . 2 (𝜑𝐴 ∈ ℝ*)
2 xrlenltd.b . 2 (𝜑𝐵 ∈ ℝ*)
3 xrlenlt 11326 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
41, 2, 3syl2anc 584 1 (𝜑 → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wcel 2108   class class class wbr 5143  *cxr 11294   < clt 11295  cle 11296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-xp 5691  df-cnv 5693  df-le 11301
This theorem is referenced by:  xrnltled  11329  supxrleub  13368  infxrgelb  13377  ixxub  13408  ixxlb  13409  icodisj  13516  supicclub2  13544  bldisj  24408  icombl  25599  ioorcl2  25607  ply1divmo  26175  ig1peu  26214  psercnlem1  26469  infxrge0gelb  32770  supxrgere  45344  supxrgelem  45348  lenelioc  45549  iccdificc  45552  limsupub  45719  fge0iccico  46385  sge0sn  46394  sge0rpcpnf  46436  pimltmnf2f  46712  pimconstlt0  46716  pimgtpnf2f  46720  pimdecfgtioo  46732  pimincfltioo  46733
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