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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 9443 | . 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 3929 ℝcr 7619 0cc0 7620 < clt 7800 ℝ+crp 9441 |
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 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-rab 2425 df-v 2688 df-un 3075 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-rp 9442 |
This theorem is referenced by: 1rp 9445 2rp 9446 3rp 9447 resqrexlemnm 10790 resqrexlemga 10795 epr 11488 pirp 12870 coseq0negpitopi 12917 pigt3 12925 |
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