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Mirrors > Home > ILE Home > Th. List > ge0gtmnf | 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 9353 | . 2 ⊢ -∞ < 0 | |
2 | mnfxr 7641 | . . . 4 ⊢ -∞ ∈ ℝ* | |
3 | 0xr 7631 | . . . 4 ⊢ 0 ∈ ℝ* | |
4 | xrltletr 9373 | . . . 4 ⊢ ((-∞ ∈ ℝ* ∧ 0 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → ((-∞ < 0 ∧ 0 ≤ 𝐴) → -∞ < 𝐴)) | |
5 | 2, 3, 4 | mp3an12 1270 | . . 3 ⊢ (𝐴 ∈ ℝ* → ((-∞ < 0 ∧ 0 ≤ 𝐴) → -∞ < 𝐴)) |
6 | 5 | imp 123 | . 2 ⊢ ((𝐴 ∈ ℝ* ∧ (-∞ < 0 ∧ 0 ≤ 𝐴)) → -∞ < 𝐴) |
7 | 1, 6 | mpanr1 429 | 1 ⊢ ((𝐴 ∈ ℝ* ∧ 0 ≤ 𝐴) → -∞ < 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1445 class class class wbr 3867 0cc0 7447 -∞cmnf 7617 ℝ*cxr 7618 < clt 7619 ≤ cle 7620 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1re 7536 ax-addrcl 7539 ax-rnegex 7551 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 |
This theorem depends on definitions: df-bi 116 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-po 4147 df-iso 4148 df-xp 4473 df-cnv 4475 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 |
This theorem is referenced by: ge0nemnf 9390 xrrege0 9391 |
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