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Mirrors > Home > MPE Home > Th. List > xrge0neqmnf | Structured version Visualization version GIF version |
Description: An extended nonnegative real cannot be minus infinity. (Contributed by Thierry Arnoux, 9-Jun-2017.) |
Ref | Expression |
---|---|
xrge0neqmnf | ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ≠ -∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnflt0 11835 | . . . . 5 ⊢ -∞ < 0 | |
2 | mnfxr 9975 | . . . . . 6 ⊢ -∞ ∈ ℝ* | |
3 | 0xr 9965 | . . . . . 6 ⊢ 0 ∈ ℝ* | |
4 | xrltnle 9984 | . . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ*) → (-∞ < 0 ↔ ¬ 0 ≤ -∞)) | |
5 | 2, 3, 4 | mp2an 704 | . . . . 5 ⊢ (-∞ < 0 ↔ ¬ 0 ≤ -∞) |
6 | 1, 5 | mpbi 219 | . . . 4 ⊢ ¬ 0 ≤ -∞ |
7 | simp2 1055 | . . . . 5 ⊢ ((-∞ ∈ ℝ* ∧ 0 ≤ -∞ ∧ -∞ ≤ +∞) → 0 ≤ -∞) | |
8 | 7 | con3i 149 | . . . 4 ⊢ (¬ 0 ≤ -∞ → ¬ (-∞ ∈ ℝ* ∧ 0 ≤ -∞ ∧ -∞ ≤ +∞)) |
9 | pnfxr 9971 | . . . . . . 7 ⊢ +∞ ∈ ℝ* | |
10 | elicc1 12090 | . . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ*) → (-∞ ∈ (0[,]+∞) ↔ (-∞ ∈ ℝ* ∧ 0 ≤ -∞ ∧ -∞ ≤ +∞))) | |
11 | 3, 9, 10 | mp2an 704 | . . . . . 6 ⊢ (-∞ ∈ (0[,]+∞) ↔ (-∞ ∈ ℝ* ∧ 0 ≤ -∞ ∧ -∞ ≤ +∞)) |
12 | 11 | biimpi 205 | . . . . 5 ⊢ (-∞ ∈ (0[,]+∞) → (-∞ ∈ ℝ* ∧ 0 ≤ -∞ ∧ -∞ ≤ +∞)) |
13 | 12 | con3i 149 | . . . 4 ⊢ (¬ (-∞ ∈ ℝ* ∧ 0 ≤ -∞ ∧ -∞ ≤ +∞) → ¬ -∞ ∈ (0[,]+∞)) |
14 | 6, 8, 13 | mp2b 10 | . . 3 ⊢ ¬ -∞ ∈ (0[,]+∞) |
15 | nelneq 2712 | . . 3 ⊢ ((𝐴 ∈ (0[,]+∞) ∧ ¬ -∞ ∈ (0[,]+∞)) → ¬ 𝐴 = -∞) | |
16 | 14, 15 | mpan2 703 | . 2 ⊢ (𝐴 ∈ (0[,]+∞) → ¬ 𝐴 = -∞) |
17 | 16 | neqned 2789 | 1 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ≠ -∞) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 class class class wbr 4583 (class class class)co 6549 0cc0 9815 +∞cpnf 9950 -∞cmnf 9951 ℝ*cxr 9952 < clt 9953 ≤ cle 9954 [,]cicc 12049 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-i2m1 9883 ax-1ne0 9884 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-icc 12053 |
This theorem is referenced by: xrge0nre 12148 xrge0adddir 29023 xrge0npcan 29025 hasheuni 29474 esumcvgre 29480 carsgclctunlem2 29708 sge0nemnf 39313 |
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