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Mirrors > Home > MPE Home > Th. List > ge0gtmnf | Structured version Visualization version GIF version |
Description: A nonnegative extended real is greater than negative infinity. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
ge0gtmnf | ⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → -∞ < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mnflt0 12603 | . 2 ⊢ -∞ < 0 | |
2 | mnfxr 10776 | . . . 4 ⊢ -∞ ∈ ℝ* | |
3 | 0xr 10766 | . . . 4 ⊢ 0 ∈ ℝ* | |
4 | xrltletr 12633 | . . . 4 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → ((-∞ < 0 ∧ 0 ≤ 𝐴) → -∞ < 𝐴)) | |
5 | 2, 3, 4 | mp3an12 1452 | . . 3 ⊢ (𝐴 ∈ ℝ* → ((-∞ < 0 ∧ 0 ≤ 𝐴) → -∞ < 𝐴)) |
6 | 5 | imp 410 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ (-∞ < 0 ∧ 0 ≤ 𝐴)) → -∞ < 𝐴) |
7 | 1, 6 | mpanr1 703 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → -∞ < 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∈ wcel 2114 class class class wbr 5030 0cc0 10615 -∞cmnf 10751 ℝ*cxr 10752 < clt 10753 ≤ cle 10754 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-addrcl 10676 ax-rnegex 10686 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-op 4523 df-uni 4797 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5429 df-po 5442 df-so 5443 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 |
This theorem is referenced by: ge0nemnf 12649 xrrege0 12650 pcgcd1 16313 pnfnei 21971 isnghm3 23478 ovolunnul 24252 radcnvle 25167 psercnlem1 25172 |
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