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Mirrors > Home > ILE Home > Th. List > elicc2 | Unicode version |
Description: Membership in a closed real interval. (Contributed by Paul Chapman, 21-Sep-2007.) (Revised by Mario Carneiro, 14-Jun-2014.) |
Ref | Expression |
---|---|
elicc2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexr 7936 | . . 3 | |
2 | rexr 7936 | . . 3 | |
3 | elicc1 9852 | . . 3 | |
4 | 1, 2, 3 | syl2an 287 | . 2 |
5 | mnfxr 7947 | . . . . . . . 8 | |
6 | 5 | a1i 9 | . . . . . . 7 |
7 | 1 | ad2antrr 480 | . . . . . . 7 |
8 | simpr1 992 | . . . . . . 7 | |
9 | mnflt 9711 | . . . . . . . 8 | |
10 | 9 | ad2antrr 480 | . . . . . . 7 |
11 | simpr2 993 | . . . . . . 7 | |
12 | 6, 7, 8, 10, 11 | xrltletrd 9739 | . . . . . 6 |
13 | 2 | ad2antlr 481 | . . . . . . 7 |
14 | pnfxr 7943 | . . . . . . . 8 | |
15 | 14 | a1i 9 | . . . . . . 7 |
16 | simpr3 994 | . . . . . . 7 | |
17 | ltpnf 9708 | . . . . . . . 8 | |
18 | 17 | ad2antlr 481 | . . . . . . 7 |
19 | 8, 13, 15, 16, 18 | xrlelttrd 9738 | . . . . . 6 |
20 | xrrebnd 9747 | . . . . . . 7 | |
21 | 8, 20 | syl 14 | . . . . . 6 |
22 | 12, 19, 21 | mpbir2and 933 | . . . . 5 |
23 | 22, 11, 16 | 3jca 1166 | . . . 4 |
24 | 23 | ex 114 | . . 3 |
25 | rexr 7936 | . . . 4 | |
26 | 25 | 3anim1i 1174 | . . 3 |
27 | 24, 26 | impbid1 141 | . 2 |
28 | 4, 27 | 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 cicc 9819 |
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-icc 9823 |
This theorem is referenced by: elicc2i 9867 iccssre 9883 iccsupr 9894 iccneg 9917 iccshftr 9922 iccshftl 9924 iccdil 9926 icccntr 9928 iccf1o 9932 suplociccreex 13160 suplociccex 13161 ivthinclemlopn 13172 ivthinclemuopn 13174 |
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