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Theorem xrnmnfpnf 39076
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 554 . . 3 (𝜑 → (𝐴 ∈ ℝ*𝐴 ≠ -∞))
4 xrnemnf 11936 . . 3 ((𝐴 ∈ ℝ*𝐴 ≠ -∞) ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
53, 4sylib 208 . 2 (𝜑 → (𝐴 ∈ ℝ ∨ 𝐴 = +∞))
6 xrnmnfpnf.2 . 2 (𝜑 → ¬ 𝐴 ∈ ℝ)
7 pm2.53 388 . 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  wo 383  wa 384   = wceq 1481  wcel 1988  wne 2791  cr 9920  +∞cpnf 10056  -∞cmnf 10057  *cxr 10058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-er 7727  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063
This theorem is referenced by:  infxr  39396
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