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Mirrors > Home > ILE Home > Th. List > mnfnre | GIF version |
Description: Minus infinity is not a real number. (Contributed by NM, 13-Oct-2005.) |
Ref | Expression |
---|---|
mnfnre | ⊢ -∞ ∉ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnex 6803 | . . . . 5 ⊢ ℂ ∈ V | |
2 | 2pwuninelg 5839 | . . . . 5 ⊢ (ℂ ∈ V → ¬ 𝒫 𝒫 ∪ ℂ ∈ ℂ) | |
3 | 1, 2 | ax-mp 7 | . . . 4 ⊢ ¬ 𝒫 𝒫 ∪ ℂ ∈ ℂ |
4 | df-mnf 6860 | . . . . . 6 ⊢ -∞ = 𝒫 +∞ | |
5 | df-pnf 6859 | . . . . . . 7 ⊢ +∞ = 𝒫 ∪ ℂ | |
6 | 5 | pweqi 3355 | . . . . . 6 ⊢ 𝒫 +∞ = 𝒫 𝒫 ∪ ℂ |
7 | 4, 6 | eqtri 2057 | . . . . 5 ⊢ -∞ = 𝒫 𝒫 ∪ ℂ |
8 | 7 | eleq1i 2100 | . . . 4 ⊢ (-∞ ∈ ℂ ↔ 𝒫 𝒫 ∪ ℂ ∈ ℂ) |
9 | 3, 8 | mtbir 595 | . . 3 ⊢ ¬ -∞ ∈ ℂ |
10 | recn 6812 | . . 3 ⊢ (-∞ ∈ ℝ → -∞ ∈ ℂ) | |
11 | 9, 10 | mto 587 | . 2 ⊢ ¬ -∞ ∈ ℝ |
12 | 11 | nelir 2294 | 1 ⊢ -∞ ∉ ℝ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 1390 ∉ wnel 2202 Vcvv 2551 𝒫 cpw 3351 ∪ cuni 3571 ℂcc 6709 ℝcr 6710 +∞cpnf 6854 -∞cmnf 6855 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-setind 4220 ax-cnex 6774 ax-resscn 6775 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-nel 2204 df-ral 2305 df-v 2553 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-uni 3572 df-pnf 6859 df-mnf 6860 |
This theorem is referenced by: renemnf 6871 xrltnr 8471 nltmnf 8479 |
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