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Theorem pnfged 42281
 Description: Plus infinity is an upper bound for extended reals. (Contributed by Glauco Siliprandi, 5-Feb-2022.)
Hypothesis
Ref Expression
pnfged.1 (𝜑𝐴 ∈ ℝ*)
Assertion
Ref Expression
pnfged (𝜑𝐴 ≤ +∞)

Proof of Theorem pnfged
StepHypRef Expression
1 pnfged.1 . 2 (𝜑𝐴 ∈ ℝ*)
2 pnfge 12533 . 2 (𝐴 ∈ ℝ*𝐴 ≤ +∞)
31, 2syl 17 1 (𝜑𝐴 ≤ +∞)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2111   class class class wbr 5034  +∞cpnf 10679  ℝ*cxr 10681   ≤ cle 10683 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pow 5235  ax-pr 5299  ax-un 7454  ax-cnex 10600  ax-resscn 10601 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-br 5035  df-opab 5097  df-xp 5529  df-cnv 5531  df-pnf 10684  df-mnf 10685  df-xr 10686  df-ltxr 10687  df-le 10688 This theorem is referenced by:  xlimpnfvlem2  42647  xlimliminflimsup  42672
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