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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 8403 | . . . 4 ⊢ ¬ 𝒫 𝒫 ∪ ℂ ∈ ℂ | |
2 | df-mnf 10414 | . . . . . 6 ⊢ -∞ = 𝒫 +∞ | |
3 | df-pnf 10413 | . . . . . . 7 ⊢ +∞ = 𝒫 ∪ ℂ | |
4 | 3 | pweqi 4383 | . . . . . 6 ⊢ 𝒫 +∞ = 𝒫 𝒫 ∪ ℂ |
5 | 2, 4 | eqtri 2802 | . . . . 5 ⊢ -∞ = 𝒫 𝒫 ∪ ℂ |
6 | 5 | eleq1i 2850 | . . . 4 ⊢ (-∞ ∈ ℂ ↔ 𝒫 𝒫 ∪ ℂ ∈ ℂ) |
7 | 1, 6 | mtbir 315 | . . 3 ⊢ ¬ -∞ ∈ ℂ |
8 | recn 10362 | . . 3 ⊢ (-∞ ∈ ℝ → -∞ ∈ ℂ) | |
9 | 7, 8 | mto 189 | . 2 ⊢ ¬ -∞ ∈ ℝ |
10 | 9 | nelir 3078 | 1 ⊢ -∞ ∉ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2107 ∉ wnel 3075 𝒫 cpw 4379 ∪ cuni 4671 ℂcc 10270 ℝcr 10271 +∞cpnf 10408 -∞cmnf 10409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-resscn 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-pnf 10413 df-mnf 10414 |
This theorem is referenced by: renemnf 10425 ltxrlt 10447 xrltnr 12264 nltmnf 12274 hashnemnf 13449 mnfnei 21433 deg1nn0clb 24287 mnfnre2 40529 |
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