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Mirrors > Home > ILE Home > Th. List > 0xr | GIF version |
Description: Zero is an extended real. (Contributed by Mario Carneiro, 15-Jun-2014.) |
Ref | Expression |
---|---|
0xr | ⊢ 0 ∈ ℝ* |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressxr 7777 | . 2 ⊢ ℝ ⊆ ℝ* | |
2 | 0re 7734 | . 2 ⊢ 0 ∈ ℝ | |
3 | 1, 2 | sselii 3064 | 1 ⊢ 0 ∈ ℝ* |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 ℝcr 7587 0cc0 7588 ℝ*cxr 7767 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-1re 7682 ax-addrcl 7685 ax-rnegex 7697 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-xr 7772 |
This theorem is referenced by: 0lepnf 9544 ge0gtmnf 9574 xlt0neg1 9589 xlt0neg2 9590 xle0neg1 9591 xle0neg2 9592 xaddf 9595 xaddval 9596 xaddid1 9613 xaddid2 9614 xnn0xadd0 9618 xaddge0 9629 xsubge0 9632 xposdif 9633 ioopos 9701 elxrge0 9729 0e0iccpnf 9731 xrmaxadd 10998 xrminrpcl 11011 xrbdtri 11013 ef01bndlem 11390 sin01bnd 11391 cos01bnd 11392 cos1bnd 11393 sin01gt0 11395 cos01gt0 11396 sin02gt0 11397 sincos1sgn 11398 sincos2sgn 11399 cos12dec 11401 halfleoddlt 11518 psmetge0 12427 isxmet2d 12444 xmetge0 12461 blgt0 12498 xblss2ps 12500 xblss2 12501 xblm 12513 bdxmet 12597 bdmet 12598 bdmopn 12600 xmetxp 12603 cnblcld 12631 blssioo 12641 sin0pilem1 12789 sin0pilem2 12790 pilem3 12791 sinhalfpilem 12799 sincosq1lem 12833 sincosq1sgn 12834 sincosq2sgn 12835 sinq12gt0 12838 cosq14gt0 12840 tangtx 12846 sincos4thpi 12848 pigt3 12852 cosordlem 12857 cosq34lt1 12858 cos02pilt1 12859 taupi 13166 |
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