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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 7804 | . . 3 | |
2 | rexr 7804 | . . 3 | |
3 | elicc1 9700 | . . 3 | |
4 | 1, 2, 3 | syl2an 287 | . 2 |
5 | mnfxr 7815 | . . . . . . . 8 | |
6 | 5 | a1i 9 | . . . . . . 7 |
7 | 1 | ad2antrr 479 | . . . . . . 7 |
8 | simpr1 987 | . . . . . . 7 | |
9 | mnflt 9562 | . . . . . . . 8 | |
10 | 9 | ad2antrr 479 | . . . . . . 7 |
11 | simpr2 988 | . . . . . . 7 | |
12 | 6, 7, 8, 10, 11 | xrltletrd 9587 | . . . . . 6 |
13 | 2 | ad2antlr 480 | . . . . . . 7 |
14 | pnfxr 7811 | . . . . . . . 8 | |
15 | 14 | a1i 9 | . . . . . . 7 |
16 | simpr3 989 | . . . . . . 7 | |
17 | ltpnf 9560 | . . . . . . . 8 | |
18 | 17 | ad2antlr 480 | . . . . . . 7 |
19 | 8, 13, 15, 16, 18 | xrlelttrd 9586 | . . . . . 6 |
20 | xrrebnd 9595 | . . . . . . 7 | |
21 | 8, 20 | syl 14 | . . . . . 6 |
22 | 12, 19, 21 | mpbir2and 928 | . . . . 5 |
23 | 22, 11, 16 | 3jca 1161 | . . . 4 |
24 | 23 | ex 114 | . . 3 |
25 | rexr 7804 | . . . 4 | |
26 | 25 | 3anim1i 1167 | . . 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 962 wcel 1480 class class class wbr 3924 (class class class)co 5767 cr 7612 cpnf 7790 cmnf 7791 cxr 7792 clt 7793 cle 7794 cicc 9667 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 |
This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-icc 9671 |
This theorem is referenced by: elicc2i 9715 iccssre 9731 iccsupr 9742 iccneg 9765 iccshftr 9770 iccshftl 9772 iccdil 9774 icccntr 9776 iccf1o 9780 suplociccreex 12760 suplociccex 12761 ivthinclemlopn 12772 ivthinclemuopn 12774 |
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