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Theorem pnfged 44669
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 13107 . 2 (𝐴 ∈ ℝ*𝐴 ≤ +∞)
31, 2syl 17 1 (𝜑𝐴 ≤ +∞)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2098   class class class wbr 5138  +∞cpnf 11242  *cxr 11244  cle 11246
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-xp 5672  df-cnv 5674  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251
This theorem is referenced by:  xlimpnfvlem2  45038  xlimliminflimsup  45063  pimgtpnf2f  45906
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