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Theorem xgepnf 9715
Description: An extended real which is greater than plus infinity is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
Assertion
Ref Expression
xgepnf (𝐴 ∈ ℝ* → (+∞ ≤ 𝐴𝐴 = +∞))

Proof of Theorem xgepnf
StepHypRef Expression
1 pnfxr 7925 . . 3 +∞ ∈ ℝ*
2 xrlenlt 7937 . . 3 ((+∞ ∈ ℝ*𝐴 ∈ ℝ*) → (+∞ ≤ 𝐴 ↔ ¬ 𝐴 < +∞))
31, 2mpan 421 . 2 (𝐴 ∈ ℝ* → (+∞ ≤ 𝐴 ↔ ¬ 𝐴 < +∞))
4 nltpnft 9713 . 2 (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
53, 4bitr4d 190 1 (𝐴 ∈ ℝ* → (+∞ ≤ 𝐴𝐴 = +∞))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 104   = wceq 1335  wcel 2128   class class class wbr 3965  +∞cpnf 7904  *cxr 7906   < clt 7907  cle 7908
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4495  ax-cnex 7818  ax-resscn 7819  ax-pre-ltirr 7839
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4591  df-cnv 4593  df-pnf 7909  df-mnf 7910  df-xr 7911  df-ltxr 7912  df-le 7913
This theorem is referenced by:  xnn0lenn0nn0  9764  xleaddadd  9786
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