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Mirrors > Home > ILE Home > Th. List > eqinfti | Unicode version |
Description: Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.) |
Ref | Expression |
---|---|
eqinfti.ti |
Ref | Expression |
---|---|
eqinfti | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inf 6872 | . . 3 inf | |
2 | eqinfti.ti | . . . . . 6 | |
3 | 2 | cnvti 6906 | . . . . 5 |
4 | 3 | eqsupti 6883 | . . . 4 |
5 | vex 2689 | . . . . . . . . . . 11 | |
6 | brcnvg 4720 | . . . . . . . . . . . 12 | |
7 | 6 | bicomd 140 | . . . . . . . . . . 11 |
8 | 5, 7 | mpan2 421 | . . . . . . . . . 10 |
9 | 8 | notbid 656 | . . . . . . . . 9 |
10 | 9 | ralbidv 2437 | . . . . . . . 8 |
11 | brcnvg 4720 | . . . . . . . . . . . 12 | |
12 | 5, 11 | mpan 420 | . . . . . . . . . . 11 |
13 | 12 | bicomd 140 | . . . . . . . . . 10 |
14 | vex 2689 | . . . . . . . . . . . . . 14 | |
15 | 5, 14 | brcnv 4722 | . . . . . . . . . . . . 13 |
16 | 15 | a1i 9 | . . . . . . . . . . . 12 |
17 | 16 | bicomd 140 | . . . . . . . . . . 11 |
18 | 17 | rexbidv 2438 | . . . . . . . . . 10 |
19 | 13, 18 | imbi12d 233 | . . . . . . . . 9 |
20 | 19 | ralbidv 2437 | . . . . . . . 8 |
21 | 10, 20 | anbi12d 464 | . . . . . . 7 |
22 | 21 | pm5.32i 449 | . . . . . 6 |
23 | 3anass 966 | . . . . . 6 | |
24 | 3anass 966 | . . . . . 6 | |
25 | 22, 23, 24 | 3bitr4i 211 | . . . . 5 |
26 | 25 | biimpi 119 | . . . 4 |
27 | 4, 26 | impel 278 | . . 3 |
28 | 1, 27 | syl5eq 2184 | . 2 inf |
29 | 28 | ex 114 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3a 962 wceq 1331 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: eqinftid 6908 |
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