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Mirrors > Home > ILE Home > Th. List > elrp | GIF version |
Description: Membership in the set of positive reals. (Contributed by NM, 27-Oct-2007.) |
Ref | Expression |
---|---|
elrp | ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 3768 | . 2 ⊢ (𝑥 = 𝐴 → (0 < 𝑥 ↔ 0 < 𝐴)) | |
2 | df-rp 8584 | . 2 ⊢ ℝ+ = {𝑥 ∈ ℝ ∣ 0 < 𝑥} | |
3 | 1, 2 | elrab2 2700 | 1 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 ↔ wb 98 ∈ 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: elrpii 8586 nnrp 8592 rpgt0 8594 rpregt0 8596 ralrp 8604 rexrp 8605 rpaddcl 8606 rpmulcl 8607 rpdivcl 8608 rpgecl 8611 rphalflt 8612 ge0p1rp 8614 rpnegap 8615 ltsubrp 8617 ltaddrp 8618 difrp 8619 elrpd 8620 iccdil 8866 icccntr 8868 expgt0 9288 sqrtdiv 9640 mulcn2 9833 |
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