Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mnfnre | Structured version Visualization version 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 | 2pwuninel 8000 | . . . 4 ⊢ ¬ 𝒫 𝒫 ∪ ℂ ∈ ℂ | |
2 | df-mnf 9956 | . . . . . 6 ⊢ -∞ = 𝒫 +∞ | |
3 | df-pnf 9955 | . . . . . . 7 ⊢ +∞ = 𝒫 ∪ ℂ | |
4 | 3 | pweqi 4112 | . . . . . 6 ⊢ 𝒫 +∞ = 𝒫 𝒫 ∪ ℂ |
5 | 2, 4 | eqtri 2632 | . . . . 5 ⊢ -∞ = 𝒫 𝒫 ∪ ℂ |
6 | 5 | eleq1i 2679 | . . . 4 ⊢ (-∞ ∈ ℂ ↔ 𝒫 𝒫 ∪ ℂ ∈ ℂ) |
7 | 1, 6 | mtbir 312 | . . 3 ⊢ ¬ -∞ ∈ ℂ |
8 | recn 9905 | . . 3 ⊢ (-∞ ∈ ℝ → -∞ ∈ ℂ) | |
9 | 7, 8 | mto 187 | . 2 ⊢ ¬ -∞ ∈ ℝ |
10 | 9 | nelir 2886 | 1 ⊢ -∞ ∉ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 ∉ wnel 2781 𝒫 cpw 4108 ∪ cuni 4372 ℂcc 9813 ℝcr 9814 +∞cpnf 9950 -∞cmnf 9951 |
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-resscn 9872 |
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-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 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-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 |
This theorem is referenced by: renemnf 9967 ltxrlt 9987 xrltnr 11829 nltmnf 11839 hashnemnf 12994 mnfnei 20835 deg1nn0clb 23654 |
Copyright terms: Public domain | W3C validator |