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Mirrors > Home > ILE Home > Th. List > rpred | GIF version |
Description: A positive real is a real. (Contributed by Mario Carneiro, 28-May-2016.) |
Ref | Expression |
---|---|
rpred.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
Ref | Expression |
---|---|
rpred | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpssre 8593 | . 2 ⊢ ℝ+ ⊆ ℝ | |
2 | rpred.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
3 | 1, 2 | sseldi 2943 | 1 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 ℝcr 6888 ℝ+crp 8583 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-rab 2315 df-in 2924 df-ss 2931 df-rp 8584 |
This theorem is referenced by: rpxrd 8623 rpcnd 8624 rpregt0d 8629 rprege0d 8630 rprene0d 8631 rprecred 8634 ltmulgt11d 8658 ltmulgt12d 8659 gt0divd 8660 ge0divd 8661 lediv12ad 8682 ltexp2a 9306 leexp2a 9307 expnlbnd2 9374 cvg1nlemcxze 9581 cvg1nlemcau 9583 cvg1nlemres 9584 cvg1n 9585 resqrexlemp1rp 9604 resqrexlemfp1 9607 resqrexlemover 9608 resqrexlemdec 9609 resqrexlemdecn 9610 resqrexlemlo 9611 resqrexlemcalc1 9612 resqrexlemcalc2 9613 resqrexlemcalc3 9614 resqrexlemnmsq 9615 resqrexlemnm 9616 resqrexlemcvg 9617 resqrexlemgt0 9618 resqrexlemoverl 9619 resqrexlemglsq 9620 resqrexlemga 9621 cau3lem 9710 addcn2 9831 mulcn2 9833 climrecvg1n 9867 climcvg1nlem 9868 qdencn 10124 |
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