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Mirrors > Home > ILE Home > Th. List > infglbti | Unicode version |
Description: An infimum is the greatest lower bound. See also infclti 6996 and inflbti 6997. (Contributed by Jim Kingdon, 18-Dec-2021.) |
Ref | Expression |
---|---|
infclti.ti | |
infclti.ex |
Ref | Expression |
---|---|
infglbti | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inf 6958 | . . . . 5 inf | |
2 | 1 | breq1i 3994 | . . . 4 inf |
3 | simpr 109 | . . . . 5 | |
4 | infclti.ti | . . . . . . . 8 | |
5 | 4 | cnvti 6992 | . . . . . . 7 |
6 | infclti.ex | . . . . . . . 8 | |
7 | 6 | cnvinfex 6991 | . . . . . . 7 |
8 | 5, 7 | supclti 6971 | . . . . . 6 |
9 | 8 | adantr 274 | . . . . 5 |
10 | brcnvg 4790 | . . . . . 6 | |
11 | 10 | bicomd 140 | . . . . 5 |
12 | 3, 9, 11 | syl2anc 409 | . . . 4 |
13 | 2, 12 | syl5bb 191 | . . 3 inf |
14 | 5, 7 | suplubti 6973 | . . . . 5 |
15 | 14 | expdimp 257 | . . . 4 |
16 | vex 2733 | . . . . . 6 | |
17 | brcnvg 4790 | . . . . . 6 | |
18 | 3, 16, 17 | sylancl 411 | . . . . 5 |
19 | 18 | rexbidv 2471 | . . . 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 2141 wral 2448 wrex 2449 cvv 2730 class class class wbr 3987 ccnv 4608 csup 6955 infcinf 6956 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-14 2144 ax-ext 2152 ax-sep 4105 ax-pow 4158 ax-pr 4192 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-un 3125 df-in 3127 df-ss 3134 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-br 3988 df-opab 4049 df-cnv 4617 df-iota 5158 df-riota 5806 df-sup 6957 df-inf 6958 |
This theorem is referenced by: infnlbti 6999 zssinfcl 11890 |
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