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Mirrors > Home > ILE Home > Th. List > lbicc2 | Unicode version |
Description: The lower bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.) (Revised by Mario Carneiro, 9-Sep-2015.) |
Ref | Expression |
---|---|
lbicc2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 992 | . 2 | |
2 | xrleid 9744 | . . 3 | |
3 | 2 | 3ad2ant1 1013 | . 2 |
4 | simp3 994 | . 2 | |
5 | elicc1 9868 | . . 3 | |
6 | 5 | 3adant3 1012 | . 2 |
7 | 1, 3, 4, 6 | mpbir3and 1175 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wb 104 w3a 973 wcel 2141 class class class wbr 3987 (class class class)co 5850 cxr 7940 cle 7942 cicc 9835 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-cnex 7852 ax-resscn 7853 ax-pre-ltirr 7873 |
This theorem depends on definitions: df-bi 116 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-id 4276 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-iota 5158 df-fun 5198 df-fv 5204 df-ov 5853 df-oprab 5854 df-mpo 5855 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-icc 9839 |
This theorem is referenced by: ivthinclemlm 13327 ivthinclemuopn 13331 ivthdec 13337 cosq34lt1 13486 cos02pilt1 13487 |
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