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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrge0nemnfd | Structured version Visualization version GIF version |
Description: A nonnegative extended real is not minus infinity. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
xrge0nemnfd.1 | ⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) |
Ref | Expression |
---|---|
xrge0nemnfd | ⊢ (𝜑 → 𝐴 ≠ -∞) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnfxr 10700 | . . 3 ⊢ -∞ ∈ ℝ* | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → -∞ ∈ ℝ*) |
3 | iccssxr 12822 | . . 3 ⊢ (0[,]+∞) ⊆ ℝ* | |
4 | xrge0nemnfd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (0[,]+∞)) | |
5 | 3, 4 | sseldi 3967 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ*) |
6 | 0xr 10690 | . . . 4 ⊢ 0 ∈ ℝ* | |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ ℝ*) |
8 | mnflt0 12523 | . . . 4 ⊢ -∞ < 0 | |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → -∞ < 0) |
10 | pnfxr 10697 | . . . . 5 ⊢ +∞ ∈ ℝ* | |
11 | 10 | a1i 11 | . . . 4 ⊢ (𝜑 → +∞ ∈ ℝ*) |
12 | iccgelb 12796 | . . . 4 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝐴 ∈ (0[,]+∞)) → 0 ≤ 𝐴) | |
13 | 7, 11, 4, 12 | syl3anc 1367 | . . 3 ⊢ (𝜑 → 0 ≤ 𝐴) |
14 | 2, 7, 5, 9, 13 | xrltletrd 12557 | . 2 ⊢ (𝜑 → -∞ < 𝐴) |
15 | 2, 5, 14 | xrgtned 41597 | 1 ⊢ (𝜑 → 𝐴 ≠ -∞) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 (class class class)co 7158 0cc0 10539 +∞cpnf 10674 -∞cmnf 10675 ℝ*cxr 10676 < clt 10677 ≤ cle 10678 [,]cicc 12744 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-addrcl 10600 ax-rnegex 10610 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-icc 12748 |
This theorem is referenced by: ovolsplit 42280 caragenuncllem 42801 |
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