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Mirrors > Home > ILE Home > Th. List > infglbti | Unicode version |
Description: An infimum is the greatest lower bound. See also infclti 6910 and inflbti 6911. (Contributed by Jim Kingdon, 18-Dec-2021.) |
Ref | Expression |
---|---|
infclti.ti | |
infclti.ex |
Ref | Expression |
---|---|
infglbti | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inf 6872 | . . . . 5 inf | |
2 | 1 | breq1i 3936 | . . . 4 inf |
3 | simpr 109 | . . . . 5 | |
4 | infclti.ti | . . . . . . . 8 | |
5 | 4 | cnvti 6906 | . . . . . . 7 |
6 | infclti.ex | . . . . . . . 8 | |
7 | 6 | cnvinfex 6905 | . . . . . . 7 |
8 | 5, 7 | supclti 6885 | . . . . . 6 |
9 | 8 | adantr 274 | . . . . 5 |
10 | brcnvg 4720 | . . . . . 6 | |
11 | 10 | bicomd 140 | . . . . 5 |
12 | 3, 9, 11 | syl2anc 408 | . . . 4 |
13 | 2, 12 | syl5bb 191 | . . 3 inf |
14 | 5, 7 | suplubti 6887 | . . . . 5 |
15 | 14 | expdimp 257 | . . . 4 |
16 | vex 2689 | . . . . . 6 | |
17 | brcnvg 4720 | . . . . . 6 | |
18 | 3, 16, 17 | sylancl 409 | . . . . 5 |
19 | 18 | rexbidv 2438 | . . . 4 |
20 | 15, 19 | sylibd 148 | . . 3 |
21 | 13, 20 | sylbid 149 | . 2 inf |
22 | 21 | expimpd 360 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wcel 1480 wral 2416 wrex 2417 cvv 2686 class class class wbr 3929 ccnv 4538 csup 6869 infcinf 6870 |
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-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 |
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-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 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-cnv 4547 df-iota 5088 df-riota 5730 df-sup 6871 df-inf 6872 |
This theorem is referenced by: infnlbti 6913 zssinfcl 11641 |
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