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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 3941 | . 2 ⊢ (𝑥 = 𝐴 → (0 < 𝑥 ↔ 0 < 𝐴)) | |
2 | df-rp 9471 | . 2 ⊢ ℝ+ = {𝑥 ∈ ℝ ∣ 0 < 𝑥} | |
3 | 1, 2 | elrab2 2847 | 1 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ wcel 1481 class class class wbr 3937 ℝcr 7643 0cc0 7644 < clt 7824 ℝ+crp 9470 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-rab 2426 df-v 2691 df-un 3080 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-rp 9471 |
This theorem is referenced by: elrpii 9473 nnrp 9480 rpgt0 9482 rpregt0 9484 ralrp 9492 rexrp 9493 rpaddcl 9494 rpmulcl 9495 rpdivcl 9496 rpgecl 9499 rphalflt 9500 ge0p1rp 9502 rpnegap 9503 negelrp 9504 ltsubrp 9507 ltaddrp 9508 difrp 9509 elrpd 9510 iccdil 9811 icccntr 9813 expgt0 10357 sqrtdiv 10846 mulcn2 11113 ef01bndlem 11499 |
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