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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 9687 | . 2 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
2 | 1 | simprbi 275 | 1 ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 class class class wbr 4018 ℝcr 7841 0cc0 7842 < clt 8023 ℝ+crp 9685 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-rab 2477 df-v 2754 df-un 3148 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-rp 9686 |
This theorem is referenced by: rpge0 9698 rpap0 9702 rpgecl 9714 0nrp 9721 rpgt0d 9731 addlelt 9800 rpsqrtcl 11085 rpmaxcl 11267 rpmincl 11281 xrminrpcl 11317 climconst 11333 blcntrps 14392 blcntr 14393 bdmet 14479 bdmopn 14481 reeff1o 14671 coseq00topi 14733 coseq0negpitopi 14734 |
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