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Mirrors > Home > ILE Home > Th. List > infglbti | Unicode version |
Description: An infimum is the greatest lower bound. See also infclti 6979 and inflbti 6980. (Contributed by Jim Kingdon, 18-Dec-2021.) |
Ref | Expression |
---|---|
infclti.ti | |
infclti.ex |
Ref | Expression |
---|---|
infglbti | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inf 6941 | . . . . 5 inf | |
2 | 1 | breq1i 3983 | . . . 4 inf |
3 | simpr 109 | . . . . 5 | |
4 | infclti.ti | . . . . . . . 8 | |
5 | 4 | cnvti 6975 | . . . . . . 7 |
6 | infclti.ex | . . . . . . . 8 | |
7 | 6 | cnvinfex 6974 | . . . . . . 7 |
8 | 5, 7 | supclti 6954 | . . . . . 6 |
9 | 8 | adantr 274 | . . . . 5 |
10 | brcnvg 4779 | . . . . . 6 | |
11 | 10 | bicomd 140 | . . . . 5 |
12 | 3, 9, 11 | syl2anc 409 | . . . 4 |
13 | 2, 12 | syl5bb 191 | . . 3 inf |
14 | 5, 7 | suplubti 6956 | . . . . 5 |
15 | 14 | expdimp 257 | . . . 4 |
16 | vex 2724 | . . . . . 6 | |
17 | brcnvg 4779 | . . . . . 6 | |
18 | 3, 16, 17 | sylancl 410 | . . . . 5 |
19 | 18 | rexbidv 2465 | . . . 4 |
20 | 15, 19 | sylibd 148 | . . 3 |
21 | 13, 20 | sylbid 149 | . 2 inf |
22 | 21 | expimpd 361 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wcel 2135 wral 2442 wrex 2443 cvv 2721 class class class wbr 3976 ccnv 4597 csup 6938 infcinf 6939 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-opab 4038 df-cnv 4606 df-iota 5147 df-riota 5792 df-sup 6940 df-inf 6941 |
This theorem is referenced by: infnlbti 6982 zssinfcl 11866 |
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