Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > elico2 | Unicode version |
Description: Membership in a closed-below, open-above real interval. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 14-Jun-2014.) |
Ref | Expression |
---|---|
elico2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 7936 | . . 3 | |
2 | elico1 9851 | . . 3 | |
3 | 1, 2 | sylan 281 | . 2 |
4 | mnfxr 7947 | . . . . . . . 8 | |
5 | 4 | a1i 9 | . . . . . . 7 |
6 | 1 | ad2antrr 480 | . . . . . . 7 |
7 | simpr1 992 | . . . . . . 7 | |
8 | mnflt 9711 | . . . . . . . 8 | |
9 | 8 | ad2antrr 480 | . . . . . . 7 |
10 | simpr2 993 | . . . . . . 7 | |
11 | 5, 6, 7, 9, 10 | xrltletrd 9739 | . . . . . 6 |
12 | simplr 520 | . . . . . . 7 | |
13 | pnfxr 7943 | . . . . . . . 8 | |
14 | 13 | a1i 9 | . . . . . . 7 |
15 | simpr3 994 | . . . . . . 7 | |
16 | pnfge 9717 | . . . . . . . 8 | |
17 | 16 | ad2antlr 481 | . . . . . . 7 |
18 | 7, 12, 14, 15, 17 | xrltletrd 9739 | . . . . . 6 |
19 | xrrebnd 9747 | . . . . . . 7 | |
20 | 7, 19 | syl 14 | . . . . . 6 |
21 | 11, 18, 20 | mpbir2and 933 | . . . . 5 |
22 | 21, 10, 15 | 3jca 1166 | . . . 4 |
23 | 22 | ex 114 | . . 3 |
24 | rexr 7936 | . . . 4 | |
25 | 24 | 3anim1i 1174 | . . 3 |
26 | 23, 25 | impbid1 141 | . 2 |
27 | 3, 26 | bitrd 187 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 w3a 967 wcel 2135 class class class wbr 3977 (class class class)co 5837 cr 7744 cpnf 7922 cmnf 7923 cxr 7924 clt 7925 cle 7926 cico 9818 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-cnex 7836 ax-resscn 7837 ax-pre-ltirr 7857 ax-pre-ltwlin 7858 ax-pre-lttrn 7859 |
This theorem depends on definitions: df-bi 116 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-nel 2430 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2724 df-sbc 2948 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-br 3978 df-opab 4039 df-id 4266 df-po 4269 df-iso 4270 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-iota 5148 df-fun 5185 df-fv 5191 df-ov 5840 df-oprab 5841 df-mpo 5842 df-pnf 7927 df-mnf 7928 df-xr 7929 df-ltxr 7930 df-le 7931 df-ico 9822 |
This theorem is referenced by: icossre 9882 elicopnf 9897 icoshft 9918 modqelico 10260 mulqaddmodid 10290 modqmuladdim 10293 addmodid 10298 icodiamlt 11109 fprodge0 11565 fprodge1 11567 cnbl0 13092 cosq34lt1 13329 cos02pilt1 13330 |
Copyright terms: Public domain | W3C validator |