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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 8271 | . . . 4 ⊢ ¬ 𝒫 𝒫 ∪ ℂ ∈ ℂ | |
2 | df-mnf 10279 | . . . . . 6 ⊢ -∞ = 𝒫 +∞ | |
3 | df-pnf 10278 | . . . . . . 7 ⊢ +∞ = 𝒫 ∪ ℂ | |
4 | 3 | pweqi 4301 | . . . . . 6 ⊢ 𝒫 +∞ = 𝒫 𝒫 ∪ ℂ |
5 | 2, 4 | eqtri 2793 | . . . . 5 ⊢ -∞ = 𝒫 𝒫 ∪ ℂ |
6 | 5 | eleq1i 2841 | . . . 4 ⊢ (-∞ ∈ ℂ ↔ 𝒫 𝒫 ∪ ℂ ∈ ℂ) |
7 | 1, 6 | mtbir 312 | . . 3 ⊢ ¬ -∞ ∈ ℂ |
8 | recn 10228 | . . 3 ⊢ (-∞ ∈ ℝ → -∞ ∈ ℂ) | |
9 | 7, 8 | mto 188 | . 2 ⊢ ¬ -∞ ∈ ℝ |
10 | 9 | nelir 3049 | 1 ⊢ -∞ ∉ ℝ |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2145 ∉ wnel 3046 𝒫 cpw 4297 ∪ cuni 4574 ℂcc 10136 ℝcr 10137 +∞cpnf 10273 -∞cmnf 10274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-resscn 10195 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-op 4323 df-uni 4575 df-br 4787 df-opab 4847 df-mpt 4864 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 |
This theorem is referenced by: renemnf 10290 ltxrlt 10310 xrltnr 12158 nltmnf 12168 hashnemnf 13336 mnfnei 21246 deg1nn0clb 24070 mnfnre2 40135 |
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