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Mirrors > Home > ILE Home > Th. List > rexr | GIF version |
Description: A standard real is an extended real. (Contributed by NM, 14-Oct-2005.) |
Ref | Expression |
---|---|
rexr | ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressxr 8026 | . 2 ⊢ ℝ ⊆ ℝ* | |
2 | 1 | sseli 3166 | 1 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2160 ℝcr 7835 ℝ*cxr 8016 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-ext 2171 |
This theorem depends on definitions: df-bi 117 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-xr 8021 |
This theorem is referenced by: rexri 8040 lenlt 8058 ltpnf 9805 mnflt 9808 xrltnsym 9818 xrlttr 9820 xrltso 9821 xrre 9845 xrre3 9847 xltnegi 9860 rexadd 9877 xaddnemnf 9882 xaddnepnf 9883 xaddcom 9886 xnegdi 9893 xpncan 9896 xnpcan 9897 xleadd1a 9898 xleadd1 9900 xltadd1 9901 xltadd2 9902 xsubge0 9906 xposdif 9907 elioo4g 9959 elioc2 9961 elico2 9962 elicc2 9963 iccss 9966 iooshf 9977 iooneg 10013 icoshft 10015 qbtwnxr 10283 modqmuladdim 10393 elicc4abs 11130 icodiamlt 11216 xrmaxrecl 11290 xrmaxaddlem 11295 xrminrecl 11308 bl2in 14340 blssps 14364 blss 14365 reopnap 14475 bl2ioo 14479 blssioo 14482 sincosq2sgn 14685 sincosq3sgn 14686 sincos6thpi 14700 |
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