MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xrlenltd Structured version   Visualization version   GIF version

Theorem xrlenltd 10316
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 10315 . 2 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ*) → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
41, 2, 3syl2anc 696 1 (𝜑 → (𝐴𝐵 ↔ ¬ 𝐵 < 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wcel 2139   class class class wbr 4804  *cxr 10285   < clt 10286  cle 10287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-xp 5272  df-cnv 5274  df-le 10292
This theorem is referenced by:  infxrgelb  12378  ixxlb  12410  infxrge0gelb  29861  supxrgere  40065  supxrgelem  40069  lenelioc  40284  iccdificc  40287  limsupub  40457  fge0iccico  41108  sge0sn  41117  sge0rpcpnf  41159  pimltmnf2  41435  pimconstlt0  41438  pimgtpnf2  41441  pimdecfgtioo  41451  pimincfltioo  41452
  Copyright terms: Public domain W3C validator