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Theorem xrmnfdc 9875
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 9808 . 2 (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2 renemnf 8037 . . . . . 6 (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
32neneqd 2381 . . . . 5 (𝐴 ∈ ℝ → ¬ 𝐴 = -∞)
43olcd 735 . . . 4 (𝐴 ∈ ℝ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
5 df-dc 836 . . . 4 (DECID 𝐴 = -∞ ↔ (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
64, 5sylibr 134 . . 3 (𝐴 ∈ ℝ → DECID 𝐴 = -∞)
7 pnfnemnf 8043 . . . . . . 7 +∞ ≠ -∞
87neii 2362 . . . . . 6 ¬ +∞ = -∞
9 eqeq1 2196 . . . . . 6 (𝐴 = +∞ → (𝐴 = -∞ ↔ +∞ = -∞))
108, 9mtbiri 676 . . . . 5 (𝐴 = +∞ → ¬ 𝐴 = -∞)
1110olcd 735 . . . 4 (𝐴 = +∞ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
1211, 5sylibr 134 . . 3 (𝐴 = +∞ → DECID 𝐴 = -∞)
13 orc 713 . . . 4 (𝐴 = -∞ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
1413, 5sylibr 134 . . 3 (𝐴 = -∞ → DECID 𝐴 = -∞)
156, 12, 143jaoi 1314 . 2 ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → DECID 𝐴 = -∞)
161, 15sylbi 121 1 (𝐴 ∈ ℝ*DECID 𝐴 = -∞)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wo 709  DECID wdc 835  w3o 979   = wceq 1364  wcel 2160  cr 7841  +∞cpnf 8020  -∞cmnf 8021  *cxr 8022
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-pnf 8025  df-mnf 8026  df-xr 8027
This theorem is referenced by:  xaddf  9876  xaddval  9877  xaddmnf1  9880  xaddcom  9893  xnegdi  9900  xpncan  9903  xleadd1a  9905  xsubge0  9913  xrmaxiflemcl  11288  xrmaxifle  11289  xrmaxiflemab  11290  xrmaxiflemlub  11291  xrmaxiflemcom  11292  xrmaxadd  11304
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