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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 3933 | . 2 ⊢ (𝑥 = 𝐴 → (0 < 𝑥 ↔ 0 < 𝐴)) | |
2 | df-rp 9442 | . 2 ⊢ ℝ+ = {𝑥 ∈ ℝ ∣ 0 < 𝑥} | |
3 | 1, 2 | elrab2 2843 | 1 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∈ 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: elrpii 9444 nnrp 9451 rpgt0 9453 rpregt0 9455 ralrp 9463 rexrp 9464 rpaddcl 9465 rpmulcl 9466 rpdivcl 9467 rpgecl 9470 rphalflt 9471 ge0p1rp 9473 rpnegap 9474 negelrp 9475 ltsubrp 9478 ltaddrp 9479 difrp 9480 elrpd 9481 iccdil 9781 icccntr 9783 expgt0 10326 sqrtdiv 10814 mulcn2 11081 ef01bndlem 11463 |
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