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Theorem xrnmnfpnf 42657
Description: An extended real that is neither real nor minus infinity, is plus infinity. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
xrnmnfpnf.1 (𝜑𝐴 ∈ ℝ*)
xrnmnfpnf.2 (𝜑 → ¬ 𝐴 ∈ ℝ)
xrnmnfpnf.3 (𝜑𝐴 ≠ -∞)
Assertion
Ref Expression
xrnmnfpnf (𝜑𝐴 = +∞)

Proof of Theorem xrnmnfpnf
StepHypRef Expression
1 xrnmnfpnf.1 . . . 4 (𝜑𝐴 ∈ ℝ*)
2 xrnmnfpnf.3 . . . 4 (𝜑𝐴 ≠ -∞)
31, 2jca 511 . . 3 (𝜑 → (𝐴 ∈ ℝ*𝐴 ≠ -∞))
4 xrnemnf 12881 . . 3 ((𝐴 ∈ ℝ*𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
53, 4sylib 217 . 2 (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
6 xrnmnfpnf.2 . 2 (𝜑 → ¬ 𝐴 ∈ ℝ)
7 pm2.53 847 . 2 ((𝐴 ∈ ℝ ∨ 𝐴 = +∞) → (¬ 𝐴 ∈ ℝ → 𝐴 = +∞))
85, 6, 7sylc 65 1 (𝜑𝐴 = +∞)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wo 843   = wceq 1537  wcel 2101  wne 2938  cr 10898  +∞cpnf 11034  -∞cmnf 11035  *cxr 11036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2103  ax-9 2111  ax-10 2132  ax-11 2149  ax-12 2166  ax-ext 2704  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7608  ax-cnex 10955  ax-resscn 10956
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2063  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2884  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3224  df-v 3436  df-sbc 3719  df-csb 3835  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4260  df-if 4463  df-pw 4538  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4842  df-br 5078  df-opab 5140  df-mpt 5161  df-id 5491  df-xp 5597  df-rel 5598  df-cnv 5599  df-co 5600  df-dm 5601  df-rn 5602  df-res 5603  df-ima 5604  df-iota 6399  df-fun 6449  df-fn 6450  df-f 6451  df-f1 6452  df-fo 6453  df-f1o 6454  df-fv 6455  df-er 8518  df-en 8754  df-dom 8755  df-sdom 8756  df-pnf 11039  df-mnf 11040  df-xr 11041
This theorem is referenced by:  infxr  42940  dfxlim2v  43423  xlimliminflimsup  43438
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