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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 9531 | . 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞)) | |
2 | renemnf 7782 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) | |
3 | 2 | neneqd 2306 | . . . . 5 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = -∞) |
4 | 3 | olcd 708 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞)) |
5 | df-dc 805 | . . . 4 ⊢ (DECID 𝐴 = -∞ ↔ (𝐴 = -∞ ∨ ¬ 𝐴 = -∞)) | |
6 | 4, 5 | sylibr 133 | . . 3 ⊢ (𝐴 ∈ ℝ → DECID 𝐴 = -∞) |
7 | pnfnemnf 7788 | . . . . . . 7 ⊢ +∞ ≠ -∞ | |
8 | 7 | neii 2287 | . . . . . 6 ⊢ ¬ +∞ = -∞ |
9 | eqeq1 2124 | . . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 = -∞ ↔ +∞ = -∞)) | |
10 | 8, 9 | mtbiri 649 | . . . . 5 ⊢ (𝐴 = +∞ → ¬ 𝐴 = -∞) |
11 | 10 | olcd 708 | . . . 4 ⊢ (𝐴 = +∞ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞)) |
12 | 11, 5 | sylibr 133 | . . 3 ⊢ (𝐴 = +∞ → DECID 𝐴 = -∞) |
13 | orc 686 | . . . 4 ⊢ (𝐴 = -∞ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞)) | |
14 | 13, 5 | sylibr 133 | . . 3 ⊢ (𝐴 = -∞ → DECID 𝐴 = -∞) |
15 | 6, 12, 14 | 3jaoi 1266 | . 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → DECID 𝐴 = -∞) |
16 | 1, 15 | sylbi 120 | 1 ⊢ (𝐴 ∈ ℝ* → DECID 𝐴 = -∞) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∨ wo 682 DECID wdc 804 ∨ w3o 946 = wceq 1316 ∈ wcel 1465 ℝcr 7587 +∞cpnf 7765 -∞cmnf 7766 ℝ*cxr 7767 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-pnf 7770 df-mnf 7771 df-xr 7772 |
This theorem is referenced by: xaddf 9595 xaddval 9596 xaddmnf1 9599 xaddcom 9612 xnegdi 9619 xpncan 9622 xleadd1a 9624 xsubge0 9632 xrmaxiflemcl 10982 xrmaxifle 10983 xrmaxiflemab 10984 xrmaxiflemlub 10985 xrmaxiflemcom 10986 xrmaxadd 10998 |
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