![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 9197 | . 2 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
4 | 1, 2, 3 | mpbir2an 889 | 1 ⊢ 𝐴 ∈ ℝ+ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1439 class class class wbr 3851 ℝcr 7410 0cc0 7411 < clt 7583 ℝ+crp 9195 |
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 9196 |
This theorem is referenced by: 1rp 9199 2rp 9200 resqrexlemnm 10512 resqrexlemga 10517 epr 11130 |
Copyright terms: Public domain | W3C validator |