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Mirrors > Home > ILE Home > Th. List > infglbti | Unicode version |
Description: An infimum is the greatest lower bound. See also infclti 6878 and inflbti 6879. (Contributed by Jim Kingdon, 18-Dec-2021.) |
Ref | Expression |
---|---|
infclti.ti | |
infclti.ex |
Ref | Expression |
---|---|
infglbti | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inf 6840 | . . . . 5 inf | |
2 | 1 | breq1i 3906 | . . . 4 inf |
3 | simpr 109 | . . . . 5 | |
4 | infclti.ti | . . . . . . . 8 | |
5 | 4 | cnvti 6874 | . . . . . . 7 |
6 | infclti.ex | . . . . . . . 8 | |
7 | 6 | cnvinfex 6873 | . . . . . . 7 |
8 | 5, 7 | supclti 6853 | . . . . . 6 |
9 | 8 | adantr 274 | . . . . 5 |
10 | brcnvg 4690 | . . . . . 6 | |
11 | 10 | bicomd 140 | . . . . 5 |
12 | 3, 9, 11 | syl2anc 408 | . . . 4 |
13 | 2, 12 | syl5bb 191 | . . 3 inf |
14 | 5, 7 | suplubti 6855 | . . . . 5 |
15 | 14 | expdimp 257 | . . . 4 |
16 | vex 2663 | . . . . . 6 | |
17 | brcnvg 4690 | . . . . . 6 | |
18 | 3, 16, 17 | sylancl 409 | . . . . 5 |
19 | 18 | rexbidv 2415 | . . . 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 1465 wral 2393 wrex 2394 cvv 2660 class class class wbr 3899 ccnv 4508 csup 6837 infcinf 6838 |
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 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-cnv 4517 df-iota 5058 df-riota 5698 df-sup 6839 df-inf 6840 |
This theorem is referenced by: infnlbti 6881 zssinfcl 11568 |
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