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Mirrors > Home > ILE Home > Th. List > elrpd | GIF version |
Description: Membership in the set of positive reals. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
elrpd.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
elrpd.2 | ⊢ (𝜑 → 0 < 𝐴) |
Ref | Expression |
---|---|
elrpd | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrpd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | elrpd.2 | . 2 ⊢ (𝜑 → 0 < 𝐴) | |
3 | elrp 8585 | . 2 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) | |
4 | 1, 2, 3 | sylanbrc 394 | 1 ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 class class class wbr 3764 ℝcr 6888 0cc0 6889 < clt 7060 ℝ+crp 8583 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rab 2315 df-v 2559 df-un 2922 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-rp 8584 |
This theorem is referenced by: modqval 9166 ltexp2a 9306 leexp2a 9307 expnlbnd2 9374 resqrexlem1arp 9603 resqrexlemp1rp 9604 resqrexlemcalc2 9613 resqrexlemcalc3 9614 resqrexlemgt0 9618 resqrexlemglsq 9620 rpsqrtcl 9639 absrpclap 9659 mulcn2 9833 climge0 9845 |
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