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Mirrors > Home > ILE Home > Th. List > supminfex | Unicode version |
Description: A supremum is the negation of the infimum of that set's image under negation. (Contributed by Jim Kingdon, 14-Jan-2022.) |
Ref | Expression |
---|---|
supminfex.ex | |
supminfex.ss |
Ref | Expression |
---|---|
supminfex | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supminfex.ex | . . . . 5 | |
2 | supminfex.ss | . . . . 5 | |
3 | 1, 2 | supinfneg 9383 | . . . 4 |
4 | ssrab2 3177 | . . . . 5 | |
5 | 4 | a1i 9 | . . . 4 |
6 | 3, 5 | infrenegsupex 9382 | . . 3 inf |
7 | elrabi 2832 | . . . . . . 7 | |
8 | 7 | adantl 275 | . . . . . 6 |
9 | 2 | sselda 3092 | . . . . . 6 |
10 | negeq 7948 | . . . . . . . . . 10 | |
11 | 10 | eleq1d 2206 | . . . . . . . . 9 |
12 | 11 | elrab3 2836 | . . . . . . . 8 |
13 | renegcl 8016 | . . . . . . . . 9 | |
14 | negeq 7948 | . . . . . . . . . . 11 | |
15 | 14 | eleq1d 2206 | . . . . . . . . . 10 |
16 | 15 | elrab3 2836 | . . . . . . . . 9 |
17 | 13, 16 | syl 14 | . . . . . . . 8 |
18 | recn 7746 | . . . . . . . . . 10 | |
19 | 18 | negnegd 8057 | . . . . . . . . 9 |
20 | 19 | eleq1d 2206 | . . . . . . . 8 |
21 | 12, 17, 20 | 3bitrd 213 | . . . . . . 7 |
22 | 21 | adantl 275 | . . . . . 6 |
23 | 8, 9, 22 | eqrdav 2136 | . . . . 5 |
24 | 23 | supeq1d 6867 | . . . 4 |
25 | 24 | negeqd 7950 | . . 3 |
26 | 6, 25 | eqtrd 2170 | . 2 inf |
27 | lttri3 7837 | . . . . . 6 | |
28 | 27 | adantl 275 | . . . . 5 |
29 | 28, 3 | infclti 6903 | . . . 4 inf |
30 | 29 | recnd 7787 | . . 3 inf |
31 | 28, 1 | supclti 6878 | . . . 4 |
32 | 31 | recnd 7787 | . . 3 |
33 | negcon2 8008 | . . 3 inf inf inf | |
34 | 30, 32, 33 | syl2anc 408 | . 2 inf inf |
35 | 26, 34 | mpbid 146 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2414 wrex 2415 crab 2418 wss 3066 class class class wbr 3924 csup 6862 infcinf 6863 cc 7611 cr 7612 clt 7793 cneg 7927 |
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-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-addcom 7713 ax-addass 7715 ax-distr 7717 ax-i2m1 7718 ax-0id 7721 ax-rnegex 7722 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-apti 7728 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 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-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 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-mpt 3986 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-isom 5127 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-sup 6864 df-inf 6865 df-pnf 7795 df-mnf 7796 df-ltxr 7798 df-sub 7928 df-neg 7929 |
This theorem is referenced by: (None) |
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