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Theorem pnfged 43716
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 13052 . 2 (𝐴 ∈ ℝ*𝐴 ≤ +∞)
31, 2syl 17 1 (𝜑𝐴 ≤ +∞)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107   class class class wbr 5106  +∞cpnf 11187  *cxr 11189  cle 11191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11108  ax-resscn 11109
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-cnv 5642  df-pnf 11192  df-mnf 11193  df-xr 11194  df-ltxr 11195  df-le 11196
This theorem is referenced by:  xlimpnfvlem2  44085  xlimliminflimsup  44110  pimgtpnf2f  44953
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