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Mirrors > Home > ILE Home > Th. List > ioossre | GIF version |
Description: An open interval is a set of reals. (Contributed by NM, 31-May-2007.) |
Ref | Expression |
---|---|
ioossre | ⊢ (𝐴(,)𝐵) ⊆ ℝ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elioore 9942 | . 2 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → 𝑥 ∈ ℝ) | |
2 | 1 | ssriv 3174 | 1 ⊢ (𝐴(,)𝐵) ⊆ ℝ |
Colors of variables: wff set class |
Syntax hints: ⊆ wss 3144 (class class class)co 5896 ℝcr 7840 (,)cioo 9918 |
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-in1 615 ax-in2 616 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-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7932 ax-resscn 7933 ax-pre-ltirr 7953 ax-pre-ltwlin 7954 ax-pre-lttrn 7955 |
This theorem depends on definitions: df-bi 117 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-id 4311 df-po 4314 df-iso 4315 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-ov 5899 df-oprab 5900 df-mpo 5901 df-pnf 8024 df-mnf 8025 df-xr 8026 df-ltxr 8027 df-le 8028 df-ioo 9922 |
This theorem is referenced by: ioof 10001 ioocosf1o 14735 |
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