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Mirrors > Home > ILE Home > Th. List > elioo2 | Unicode version |
Description: Membership in an open interval of extended reals. (Contributed by NM, 6-Feb-2007.) |
Ref | Expression |
---|---|
elioo2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iooval2 9666 | . . 3 | |
2 | 1 | eleq2d 2187 | . 2 |
3 | breq2 3903 | . . . . 5 | |
4 | breq1 3902 | . . . . 5 | |
5 | 3, 4 | anbi12d 464 | . . . 4 |
6 | 5 | elrab 2813 | . . 3 |
7 | 3anass 951 | . . 3 | |
8 | 6, 7 | bitr4i 186 | . 2 |
9 | 2, 8 | syl6bb 195 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 947 wceq 1316 wcel 1465 crab 2397 class class class wbr 3899 (class class class)co 5742 cr 7587 cxr 7767 clt 7768 cioo 9639 |
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-in1 588 ax-in2 589 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-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-cnex 7679 ax-resscn 7680 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-po 4188 df-iso 4189 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-ioo 9643 |
This theorem is referenced by: eliooord 9679 elioopnf 9718 elioomnf 9719 bl2ioo 12638 dedekindicc 12707 sin0pilem2 12790 pilem3 12791 sincosq1sgn 12834 sincosq2sgn 12835 sincosq3sgn 12836 sincosq4sgn 12837 sinq12gt0 12838 cosq14gt0 12840 cosq23lt0 12841 coseq0q4123 12842 coseq00topi 12843 coseq0negpitopi 12844 sincos6thpi 12850 cosordlem 12857 cos02pilt1 12859 taupi 13166 |
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