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Mirrors > Home > ILE Home > Th. List > xrmnfdc | GIF version |
Description: An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.) |
Ref | Expression |
---|---|
xrmnfdc | ⊢ (𝐴 ∈ ℝ* → DECID 𝐴 = -∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elxr 9808 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | renemnf 8037 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
3 | 2 | neneqd 2381 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = -∞) |
4 | 3 | olcd 735 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞)) |
5 | df-dc 836 | . . . 4 ⊢ (DECID 𝐴 = -∞ ↔ (𝐴 = -∞ ∨ ¬ 𝐴 = -∞)) | |
6 | 4, 5 | sylibr 134 | . . 3 ⊢ (𝐴 ∈ ℝ → DECID 𝐴 = -∞) |
7 | pnfnemnf 8043 | . . . . . . 7 ⊢ +∞ ≠ -∞ | |
8 | 7 | neii 2362 | . . . . . 6 ⊢ ¬ +∞ = -∞ |
9 | eqeq1 2196 | . . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 = -∞ ↔ +∞ = -∞)) | |
10 | 8, 9 | mtbiri 676 | . . . . 5 ⊢ (𝐴 = +∞ → ¬ 𝐴 = -∞) |
11 | 10 | olcd 735 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞)) |
12 | 11, 5 | sylibr 134 | . . 3 ⊢ (𝐴 = +∞ → DECID 𝐴 = -∞) |
13 | orc 713 | . . . 4 ⊢ (𝐴 = -∞ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞)) | |
14 | 13, 5 | sylibr 134 | . . 3 ⊢ (𝐴 = -∞ → DECID 𝐴 = -∞) |
15 | 6, 12, 14 | 3jaoi 1314 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → DECID 𝐴 = -∞) |
16 | 1, 15 | sylbi 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 ℝcr 7841 +∞cpnf 8020 -∞cmnf 8021 ℝ*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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