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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 | df-mnf 10667 | . . . . 5 ⊢ -∞ = 𝒫 +∞ | |
2 | df-pnf 10666 | . . . . . 6 ⊢ +∞ = 𝒫 ∪ ℂ | |
3 | 2 | pweqi 4515 | . . . . 5 ⊢ 𝒫 +∞ = 𝒫 𝒫 ∪ ℂ |
4 | 1, 3 | eqtri 2821 | . . . 4 ⊢ -∞ = 𝒫 𝒫 ∪ ℂ |
5 | 2pwuninel 8656 | . . . 4 ⊢ ¬ 𝒫 𝒫 ∪ ℂ ∈ ℂ | |
6 | 4, 5 | eqneltri 2883 | . . 3 ⊢ ¬ -∞ ∈ ℂ |
7 | recn 10616 | . . 3 ⊢ (-∞ ∈ ℝ → -∞ ∈ ℂ) | |
8 | 6, 7 | mto 200 | . 2 ⊢ ¬ -∞ ∈ ℝ |
9 | 8 | nelir 3094 | 1 ⊢ -∞ ∉ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 ∉ wnel 3091 𝒫 cpw 4497 ∪ cuni 4800 ℂcc 10524 ℝcr 10525 +∞cpnf 10661 -∞cmnf 10662 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-resscn 10583 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 |
This theorem is referenced by: renemnf 10679 ltxrlt 10700 xrltnr 12502 nltmnf 12512 hashnemnf 13700 mnfnei 21826 deg1nn0clb 24691 mnfnre2 42032 |
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