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Mirrors > Home > ILE Home > Th. List > rpgt0 | GIF version |
Description: A positive real is greater than zero. (Contributed by FL, 27-Dec-2007.) |
Ref | Expression |
---|---|
rpgt0 | ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrp 9436 | . 2 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
2 | 1 | simprbi 273 | 1 ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 class class class wbr 3924 ℝcr 7612 0cc0 7613 < clt 7793 ℝ+crp 9434 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rab 2423 df-v 2683 df-un 3070 df-sn 3528 df-pr 3529 df-op 3531 df-br 3925 df-rp 9435 |
This theorem is referenced by: rpge0 9447 rpap0 9451 rpgecl 9463 0nrp 9470 rpgt0d 9479 addlelt 9548 rpsqrtcl 10806 rpmaxcl 10988 rpmincl 11002 xrminrpcl 11036 climconst 11052 blcntrps 12573 blcntr 12574 bdmet 12660 bdmopn 12662 coseq00topi 12905 coseq0negpitopi 12906 |
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