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Mirrors > Home > ILE Home > Th. List > elrpii | GIF version |
Description: Membership in the set of positive reals. (Contributed by NM, 23-Feb-2008.) |
Ref | Expression |
---|---|
elrpi.1 | ⊢ 𝐴 ∈ ℝ |
elrpi.2 | ⊢ 0 < 𝐴 |
Ref | Expression |
---|---|
elrpii | ⊢ 𝐴 ∈ ℝ+ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrpi.1 | . 2 ⊢ 𝐴 ∈ ℝ | |
2 | elrpi.2 | . 2 ⊢ 0 < 𝐴 | |
3 | elrp 9436 | . 2 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
4 | 1, 2, 3 | mpbir2an 926 | 1 ⊢ 𝐴 ∈ ℝ+ |
Colors of variables: wff set class |
Syntax hints: ∈ 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: 1rp 9438 2rp 9439 3rp 9440 resqrexlemnm 10783 resqrexlemga 10788 epr 11477 pirp 12859 coseq0negpitopi 12906 pigt3 12914 |
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