Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > eqsupti | Unicode version |
Description: Sufficient condition for an element to be equal to the supremum. (Contributed by Jim Kingdon, 23-Nov-2021.) |
Ref | Expression |
---|---|
supmoti.ti |
Ref | Expression |
---|---|
eqsupti |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supmoti.ti | . . . . 5 | |
2 | 1 | adantlr 468 | . . . 4 |
3 | breq1 3932 | . . . . . . . . . 10 | |
4 | 3 | notbid 656 | . . . . . . . . 9 |
5 | 4 | ralbidv 2437 | . . . . . . . 8 |
6 | breq2 3933 | . . . . . . . . . 10 | |
7 | 6 | imbi1d 230 | . . . . . . . . 9 |
8 | 7 | ralbidv 2437 | . . . . . . . 8 |
9 | 5, 8 | anbi12d 464 | . . . . . . 7 |
10 | 9 | rspcev 2789 | . . . . . 6 |
11 | 10 | 3impb 1177 | . . . . 5 |
12 | 11 | adantl 275 | . . . 4 |
13 | 2, 12 | supval2ti 6882 | . . 3 |
14 | 3simpc 980 | . . . . 5 | |
15 | 14 | adantl 275 | . . . 4 |
16 | simpr1 987 | . . . . 5 | |
17 | 2, 12 | supeuti 6881 | . . . . 5 |
18 | 9 | riota2 5752 | . . . . 5 |
19 | 16, 17, 18 | syl2anc 408 | . . . 4 |
20 | 15, 19 | mpbid 146 | . . 3 |
21 | 13, 20 | eqtrd 2172 | . 2 |
22 | 21 | ex 114 | 1 |
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 wreu 2418 class class class wbr 3929 crio 5729 csup 6869 |
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-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
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-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-iota 5088 df-riota 5730 df-sup 6871 |
This theorem is referenced by: eqsuptid 6884 eqinfti 6907 maxabs 10981 xrmaxif 11020 bezoutlemsup 11697 suplociccex 12772 |
Copyright terms: Public domain | W3C validator |