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Theorem xrlenltd 11312
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 11311 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
41, 2, 3syl2anc 582 1 (𝜑 → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wcel 2098   class class class wbr 5149  *cxr 11279   < clt 11280  cle 11281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-xp 5684  df-cnv 5686  df-le 11286
This theorem is referenced by:  xrnltled  11314  supxrleub  13340  infxrgelb  13349  ixxub  13380  ixxlb  13381  icodisj  13488  supicclub2  13516  bldisj  24348  icombl  25537  ioorcl2  25545  ply1divmo  26116  ig1peu  26154  psercnlem1  26407  infxrge0gelb  32618  supxrgere  44853  supxrgelem  44857  lenelioc  45059  iccdificc  45062  limsupub  45230  fge0iccico  45896  sge0sn  45905  sge0rpcpnf  45947  pimltmnf2f  46223  pimconstlt0  46227  pimgtpnf2f  46231  pimdecfgtioo  46243  pimincfltioo  46244
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