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Theorem pnfged 44856
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 13143 . 2 (𝐴 ∈ ℝ*𝐴 ≤ +∞)
31, 2syl 17 1 (𝜑𝐴 ≤ +∞)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099   class class class wbr 5148  +∞cpnf 11276  *cxr 11278  cle 11280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-xp 5684  df-cnv 5686  df-pnf 11281  df-mnf 11282  df-xr 11283  df-ltxr 11284  df-le 11285
This theorem is referenced by:  xlimpnfvlem2  45225  xlimliminflimsup  45250  pimgtpnf2f  46093
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