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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 9190 | . 2 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
2 | 1 | simprbi 270 | 1 ⊢ (𝐴 ∈ ℝ+ → 0 < 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1439 class class class wbr 3851 ℝcr 7403 0cc0 7404 < clt 7576 ℝ+crp 9188 |
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-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-3an 927 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-rab 2369 df-v 2622 df-un 3004 df-sn 3456 df-pr 3457 df-op 3459 df-br 3852 df-rp 9189 |
This theorem is referenced by: rpge0 9200 rpap0 9204 rpgecl 9216 0nrp 9221 rpgt0d 9230 addlelt 9293 rpsqrtcl 10528 climconst 10732 |
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