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| Mirrors > Home > MPE Home > Th. List > elrp | Structured version Visualization version 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 5147 | . 2 ⊢ (𝑥 = 𝐴 → (0 < 𝑥 ↔ 0 < 𝐴)) | |
| 2 | df-rp 13035 | . 2 ⊢ ℝ+ = {𝑥 ∈ ℝ ∣ 0 < 𝑥} | |
| 3 | 1, 2 | elrab2 3695 | 1 ⊢ (𝐴 ∈ ℝ+ ↔ (𝐴 ∈ ℝ ∧ 0 < 𝐴)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 class class class wbr 5143 ℝcr 11154 0cc0 11155 < clt 11295 ℝ+crp 13034 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-rp 13035 |
| This theorem is referenced by: elrpii 13037 nnrp 13046 rpgt0 13047 rpregt0 13049 ralrp 13055 rexrp 13056 rpaddcl 13057 rpmulcl 13058 rpdivcl 13060 rpgecl 13063 rphalflt 13064 ge0p1rp 13066 rpneg 13067 negelrp 13068 ltsubrp 13071 ltaddrp 13072 difrp 13073 elrpd 13074 infmrp1 13386 dfrp2 13436 iccdil 13530 icccntr 13532 1mod 13943 expgt0 14136 resqrex 15289 sqrtdiv 15304 sqrtneglem 15305 mulcn2 15632 ef01bndlem 16220 sinltx 16225 met1stc 24534 met2ndci 24535 bcthlem4 25361 itg2mulc 25782 dvferm1 26023 dvne0 26050 reeff1o 26491 ellogdm 26681 cxpge0 26725 cxple2a 26741 cxpcn3lem 26790 cxpaddlelem 26794 cxpaddle 26795 atanbnd 26969 rlimcnp 27008 amgm 27034 chtub 27256 chebbnd1 27516 chto1ub 27520 pntlem3 27653 blocni 30824 rpdp2cl 32864 dp2ltc 32869 dplti 32887 dpgti 32888 dpexpp1 32890 dpmul4 32896 fdvposlt 34614 hgt750lem 34666 unbdqndv2lem2 36511 heiborlem8 37825 dvrelog2 42065 dvrelog3 42066 2xp3dxp2ge1d 42242 sqrtcvallem1 43644 wallispilem4 46083 perfectALTVlem2 47709 regt1loggt0 48457 |
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