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Theorem xgepnf 9814
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 8008 . . 3 +∞ ∈ ℝ*
2 xrlenlt 8020 . . 3 ((+∞ ∈ ℝ*𝐴 ∈ ℝ*) → (+∞ ≤ 𝐴 ↔ ¬ 𝐴 < +∞))
31, 2mpan 424 . 2 (𝐴 ∈ ℝ* → (+∞ ≤ 𝐴 ↔ ¬ 𝐴 < +∞))
4 nltpnft 9812 . 2 (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
53, 4bitr4d 191 1 (𝐴 ∈ ℝ* → (+∞ ≤ 𝐴𝐴 = +∞))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wb 105   = wceq 1353  wcel 2148   class class class wbr 4003  +∞cpnf 7987  *cxr 7989   < clt 7990  cle 7991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-pre-ltirr 7922
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-xp 4632  df-cnv 4634  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996
This theorem is referenced by:  xnn0lenn0nn0  9863  xleaddadd  9885
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