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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 3847 | . 2 ⊢ (𝑥 = 𝐴 → (0 < 𝑥 ↔ 0 < 𝐴)) | |
2 | df-rp 9125 | . 2 ⊢ ℝ+ = {𝑥 ∈ ℝ ∣ 0 < 𝑥} | |
3 | 1, 2 | elrab2 2774 | 1 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 102 ↔ wb 103 ∈ wcel 1438 class class class wbr 3843 ℝcr 7339 0cc0 7340 < clt 7512 ℝ+crp 9124 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-rab 2368 df-v 2621 df-un 3003 df-sn 3450 df-pr 3451 df-op 3453 df-br 3844 df-rp 9125 |
This theorem is referenced by: elrpii 9127 nnrp 9133 rpgt0 9135 rpregt0 9137 ralrp 9145 rexrp 9146 rpaddcl 9147 rpmulcl 9148 rpdivcl 9149 rpgecl 9152 rphalflt 9153 ge0p1rp 9155 rpnegap 9156 ltsubrp 9158 ltaddrp 9159 difrp 9160 elrpd 9161 iccdil 9405 icccntr 9407 expgt0 9976 sqrtdiv 10463 mulcn2 10688 ef01bndlem 11034 |
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