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Mirrors > Home > ILE Home > Th. List > eqinftid | 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 | |
eqinftid.2 | |
eqinftid.3 | |
eqinftid.4 |
Ref | Expression |
---|---|
eqinftid | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqinftid.2 | . 2 | |
2 | eqinftid.3 | . . 3 | |
3 | 2 | ralrimiva 2505 | . 2 |
4 | eqinftid.4 | . . . 4 | |
5 | 4 | expr 372 | . . 3 |
6 | 5 | ralrimiva 2505 | . 2 |
7 | eqinfti.ti | . . 3 | |
8 | 7 | eqinfti 6907 | . 2 inf |
9 | 1, 3, 6, 8 | mp3and 1318 | 1 inf |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 wrex 2417 class class class wbr 3929 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: infminti 6914 |
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