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Mirrors > Home > ILE Home > Th. List > rpre | GIF version |
Description: A positive real is a real. (Contributed by NM, 27-Oct-2007.) |
Ref | Expression |
---|---|
rpre | ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rp 9567 | . . 3 ⊢ ℝ+ = {𝑥 ∈ ℝ ∣ 0 < 𝑥} | |
2 | ssrab2 3213 | . . 3 ⊢ {𝑥 ∈ ℝ ∣ 0 < 𝑥} ⊆ ℝ | |
3 | 1, 2 | eqsstri 3160 | . 2 ⊢ ℝ+ ⊆ ℝ |
4 | 3 | sseli 3124 | 1 ⊢ (𝐴 ∈ ℝ+ → 𝐴 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2128 {crab 2439 class class class wbr 3967 ℝcr 7733 0cc0 7734 < clt 7914 ℝ+crp 9566 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-rab 2444 df-in 3108 df-ss 3115 df-rp 9567 |
This theorem is referenced by: rpxr 9574 rpcn 9575 rpssre 9577 rpge0 9579 rprege0 9581 rpap0 9583 rprene0 9584 rpreap0 9585 rpaddcl 9590 rpmulcl 9591 rpdivcl 9592 rpgecl 9595 ledivge1le 9639 addlelt 9681 iccdil 9908 expnlbnd 10551 caucvgre 10892 rennim 10913 rpsqrtcl 10952 qdenre 11113 rpmaxcl 11134 rpmincl 11148 xrminrpcl 11182 2clim 11209 cn1lem 11222 climsqz 11243 climsqz2 11244 climcau 11255 efgt1 11605 ef01bndlem 11664 bdmet 12972 bdmopn 12974 dveflem 13157 reeff1o 13164 logleb 13266 logrpap0b 13267 cxple3 13311 rpcxpsqrt 13312 rpcxpsqrtth 13320 dceqnconst 13701 dcapnconst 13702 |
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