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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 9390 | . . . 4 |
4 | ssrab2 3182 | . . . . 5 | |
5 | 4 | a1i 9 | . . . 4 |
6 | 3, 5 | infrenegsupex 9389 | . . 3 inf |
7 | elrabi 2837 | . . . . . . 7 | |
8 | 7 | adantl 275 | . . . . . 6 |
9 | 2 | sselda 3097 | . . . . . 6 |
10 | negeq 7955 | . . . . . . . . . 10 | |
11 | 10 | eleq1d 2208 | . . . . . . . . 9 |
12 | 11 | elrab3 2841 | . . . . . . . 8 |
13 | renegcl 8023 | . . . . . . . . 9 | |
14 | negeq 7955 | . . . . . . . . . . 11 | |
15 | 14 | eleq1d 2208 | . . . . . . . . . 10 |
16 | 15 | elrab3 2841 | . . . . . . . . 9 |
17 | 13, 16 | syl 14 | . . . . . . . 8 |
18 | recn 7753 | . . . . . . . . . 10 | |
19 | 18 | negnegd 8064 | . . . . . . . . 9 |
20 | 19 | eleq1d 2208 | . . . . . . . 8 |
21 | 12, 17, 20 | 3bitrd 213 | . . . . . . 7 |
22 | 21 | adantl 275 | . . . . . 6 |
23 | 8, 9, 22 | eqrdav 2138 | . . . . 5 |
24 | 23 | supeq1d 6874 | . . . 4 |
25 | 24 | negeqd 7957 | . . 3 |
26 | 6, 25 | eqtrd 2172 | . 2 inf |
27 | lttri3 7844 | . . . . . 6 | |
28 | 27 | adantl 275 | . . . . 5 |
29 | 28, 3 | infclti 6910 | . . . 4 inf |
30 | 29 | recnd 7794 | . . 3 inf |
31 | 28, 1 | supclti 6885 | . . . 4 |
32 | 31 | recnd 7794 | . . 3 |
33 | negcon2 8015 | . . 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 2416 wrex 2417 crab 2420 wss 3071 class class class wbr 3929 csup 6869 infcinf 6870 cc 7618 cr 7619 clt 7800 cneg 7934 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-apti 7735 ax-pre-ltadd 7736 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-isom 5132 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-sup 6871 df-inf 6872 df-pnf 7802 df-mnf 7803 df-ltxr 7805 df-sub 7935 df-neg 7936 |
This theorem is referenced by: (None) |
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