Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ge0nemnf2 | Structured version Visualization version GIF version |
Description: A nonnegative extended real is not -∞ (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
ge0nemnf2 | ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ≠ -∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eliccxr 39737 | . 2 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ∈ ℝ*) | |
2 | 0xr 10086 | . . . 4 ⊢ 0 ∈ ℝ* | |
3 | 2 | a1i 11 | . . 3 ⊢ (𝐴 ∈ (0[,]+∞) → 0 ∈ ℝ*) |
4 | pnfxr 10092 | . . . 4 ⊢ +∞ ∈ ℝ* | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝐴 ∈ (0[,]+∞) → +∞ ∈ ℝ*) |
6 | id 22 | . . 3 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ∈ (0[,]+∞)) | |
7 | iccgelb 12230 | . . 3 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 ∈ (0[,]+∞)) → 0 ≤ 𝐴) | |
8 | 3, 5, 6, 7 | syl3anc 1326 | . 2 ⊢ (𝐴 ∈ (0[,]+∞) → 0 ≤ 𝐴) |
9 | ge0nemnf 12004 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → 𝐴 ≠ -∞) | |
10 | 1, 8, 9 | syl2anc 693 | 1 ⊢ (𝐴 ∈ (0[,]+∞) → 𝐴 ≠ -∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ≠ wne 2794 class class class wbr 4653 (class class class)co 6650 0cc0 9936 +∞cpnf 10071 -∞cmnf 10072 ℝ*cxr 10073 ≤ cle 10075 [,]cicc 12178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-i2m1 10004 ax-1ne0 10005 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-icc 12182 |
This theorem is referenced by: sge0split 40626 |
Copyright terms: Public domain | W3C validator |