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Mirrors > Home > ILE Home > Th. List > rge0ssre | GIF version |
Description: Nonnegative real numbers are real numbers. (Contributed by Thierry Arnoux, 9-Sep-2018.) (Proof shortened by AV, 8-Sep-2019.) |
Ref | Expression |
---|---|
rge0ssre | ⊢ (0[,)+∞) ⊆ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrege0 8999 | . . 3 ⊢ (𝑥 ∈ (0[,)+∞) ↔ (𝑥 ∈ ℝ ∧ 0 ≤ 𝑥)) | |
2 | 1 | simplbi 268 | . 2 ⊢ (𝑥 ∈ (0[,)+∞) → 𝑥 ∈ ℝ) |
3 | 2 | ssriv 3003 | 1 ⊢ (0[,)+∞) ⊆ ℝ |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1433 ⊆ wss 2973 class class class wbr 3785 (class class class)co 5532 ℝcr 6980 0cc0 6981 +∞cpnf 7150 ≤ cle 7154 [,)cico 8913 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-1re 7070 ax-addrcl 7073 ax-rnegex 7085 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090 |
This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-ico 8917 |
This theorem is referenced by: (None) |
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