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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 7712 | . . . . 5 ⊢ ℂ ∈ V | |
2 | 2pwuninelg 6148 | . . . . 5 ⊢ (ℂ ∈ V → ¬ 𝒫 𝒫 ∪ ℂ ∈ ℂ) | |
3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ ¬ 𝒫 𝒫 ∪ ℂ ∈ ℂ |
4 | df-mnf 7771 | . . . . . 6 ⊢ -∞ = 𝒫 +∞ | |
5 | df-pnf 7770 | . . . . . . 7 ⊢ +∞ = 𝒫 ∪ ℂ | |
6 | 5 | pweqi 3484 | . . . . . 6 ⊢ 𝒫 +∞ = 𝒫 𝒫 ∪ ℂ |
7 | 4, 6 | eqtri 2138 | . . . . 5 ⊢ -∞ = 𝒫 𝒫 ∪ ℂ |
8 | 7 | eleq1i 2183 | . . . 4 ⊢ (-∞ ∈ ℂ ↔ 𝒫 𝒫 ∪ ℂ ∈ ℂ) |
9 | 3, 8 | mtbir 645 | . . 3 ⊢ ¬ -∞ ∈ ℂ |
10 | recn 7721 | . . 3 ⊢ (-∞ ∈ ℝ → -∞ ∈ ℂ) | |
11 | 9, 10 | mto 636 | . 2 ⊢ ¬ -∞ ∈ ℝ |
12 | 11 | nelir 2383 | 1 ⊢ -∞ ∉ ℝ |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 1465 ∉ wnel 2380 Vcvv 2660 𝒫 cpw 3480 ∪ cuni 3706 ℂcc 7586 ℝcr 7587 +∞cpnf 7765 -∞cmnf 7766 |
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-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-nel 2381 df-ral 2398 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 |
This theorem is referenced by: renemnf 7782 xrltnr 9534 nltmnf 9542 |
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