![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > renemnf | Structured version Visualization version GIF version |
Description: No real equals minus infinity. (Contributed by NM, 14-Oct-2005.) (Proof shortened by Andrew Salmon, 19-Nov-2011.) |
Ref | Expression |
---|---|
renemnf | ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfnre 10120 | . . . 4 ⊢ -∞ ∉ ℝ | |
2 | 1 | neli 2928 | . . 3 ⊢ ¬ -∞ ∈ ℝ |
3 | eleq1 2718 | . . 3 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ ↔ -∞ ∈ ℝ)) | |
4 | 2, 3 | mtbiri 316 | . 2 ⊢ (𝐴 = -∞ → ¬ 𝐴 ∈ ℝ) |
5 | 4 | necon2ai 2852 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ℝcr 9973 -∞cmnf 10110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-resscn 10031 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 |
This theorem is referenced by: renemnfd 10129 renfdisj 10136 xrnemnf 11989 rexneg 12080 rexadd 12101 xaddnemnf 12105 xaddcom 12109 xaddid1 12110 xnegdi 12116 xpncan 12119 xleadd1a 12121 rexmul 12139 xadddilem 12162 xrs1mnd 19832 xrs10 19833 isxmet2d 22179 imasdsf1olem 22225 xaddeq0 29646 icorempt2 33329 infrpge 39880 infleinflem1 39899 xrre4 39951 climxrre 40300 |
Copyright terms: Public domain | W3C validator |