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Theorem xrmnfdc 9467
Description: An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
Assertion
Ref Expression
xrmnfdc (𝐴 ∈ ℝ*DECID 𝐴 = -∞)

Proof of Theorem xrmnfdc
StepHypRef Expression
1 elxr 9404 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 renemnf 7686 . . . . . 6 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
32neneqd 2288 . . . . 5 (𝐴 ∈ ℝ → ¬ 𝐴 = -∞)
43olcd 694 . . . 4 (𝐴 ∈ ℝ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
5 df-dc 787 . . . 4 (DECID 𝐴 = -∞ ↔ (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
64, 5sylibr 133 . . 3 (𝐴 ∈ ℝ → DECID 𝐴 = -∞)
7 pnfnemnf 7692 . . . . . . 7 +∞ ≠ -∞
87neii 2269 . . . . . 6 ¬ +∞ = -∞
9 eqeq1 2106 . . . . . 6 (𝐴 = +∞ → (𝐴 = -∞ ↔ +∞ = -∞))
108, 9mtbiri 641 . . . . 5 (𝐴 = +∞ → ¬ 𝐴 = -∞)
1110olcd 694 . . . 4 (𝐴 = +∞ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
1211, 5sylibr 133 . . 3 (𝐴 = +∞ → DECID 𝐴 = -∞)
13 orc 674 . . . 4 (𝐴 = -∞ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
1413, 5sylibr 133 . . 3 (𝐴 = -∞ → DECID 𝐴 = -∞)
156, 12, 143jaoi 1249 . 2 ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → DECID 𝐴 = -∞)
161, 15sylbi 120 1 (𝐴 ∈ ℝ*DECID 𝐴 = -∞)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 670  DECID wdc 786  w3o 929   = wceq 1299  wcel 1448  cr 7499  +∞cpnf 7669  -∞cmnf 7670  *cxr 7671
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-uni 3684  df-pnf 7674  df-mnf 7675  df-xr 7676
This theorem is referenced by:  xaddf  9468  xaddval  9469  xaddmnf1  9472  xaddcom  9485  xnegdi  9492  xpncan  9495  xleadd1a  9497  xsubge0  9505  xrmaxiflemcl  10853  xrmaxifle  10854  xrmaxiflemab  10855  xrmaxiflemlub  10856  xrmaxiflemcom  10857  xrmaxadd  10869
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