Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > inflbti | Unicode version |
Description: An infimum is a lower bound. See also infclti 6436 and infglbti 6438. (Contributed by Jim Kingdon, 18-Dec-2021.) |
Ref | Expression |
---|---|
infclti.ti | |
infclti.ex |
Ref | Expression |
---|---|
inflbti | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infclti.ti | . . . . . 6 | |
2 | 1 | cnvti 6432 | . . . . 5 |
3 | infclti.ex | . . . . . 6 | |
4 | 3 | cnvinfex 6431 | . . . . 5 |
5 | 2, 4 | supubti 6412 | . . . 4 |
6 | 5 | imp 122 | . . 3 |
7 | df-inf 6398 | . . . . . 6 inf | |
8 | 7 | a1i 9 | . . . . 5 inf |
9 | 8 | breq2d 3797 | . . . 4 inf |
10 | 2, 4 | supclti 6411 | . . . . 5 |
11 | brcnvg 4534 | . . . . . 6 | |
12 | 11 | bicomd 139 | . . . . 5 |
13 | 10, 12 | sylan 277 | . . . 4 |
14 | 9, 13 | bitrd 186 | . . 3 inf |
15 | 6, 14 | mtbird 630 | . 2 inf |
16 | 15 | ex 113 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wceq 1284 wcel 1433 wral 2348 wrex 2349 class class class wbr 3785 ccnv 4362 csup 6395 infcinf 6396 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-reu 2355 df-rmo 2356 df-rab 2357 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-cnv 4371 df-iota 4887 df-riota 5488 df-sup 6397 df-inf 6398 |
This theorem is referenced by: zssinfcl 10344 infssuzledc 10346 |
Copyright terms: Public domain | W3C validator |