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Theorem xrlenltd 10699
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 10698 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
41, 2, 3syl2anc 586 1 (𝜑 → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wcel 2107   class class class wbr 5057  *cxr 10666   < clt 10667  cle 10668
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-xp 5554  df-cnv 5556  df-le 10673
This theorem is referenced by:  xrnltled  10701  supxrleub  12711  infxrgelb  12720  ixxub  12751  ixxlb  12752  icodisj  12854  supicclub2  12881  bldisj  23000  icombl  24157  ioorcl2  24165  ply1divmo  24721  ig1peu  24757  psercnlem1  25005  infxrge0gelb  30482  supxrgere  41585  supxrgelem  41589  lenelioc  41796  iccdificc  41799  limsupub  41969  fge0iccico  42637  sge0sn  42646  sge0rpcpnf  42688  pimltmnf2  42964  pimconstlt0  42967  pimgtpnf2  42970  pimdecfgtioo  42980  pimincfltioo  42981
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