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 460 | . . . 4 |
3 | breq1 3788 | . . . . . . . . . 10 | |
4 | 3 | notbid 624 | . . . . . . . . 9 |
5 | 4 | ralbidv 2368 | . . . . . . . 8 |
6 | breq2 3789 | . . . . . . . . . 10 | |
7 | 6 | imbi1d 229 | . . . . . . . . 9 |
8 | 7 | ralbidv 2368 | . . . . . . . 8 |
9 | 5, 8 | anbi12d 456 | . . . . . . 7 |
10 | 9 | rspcev 2701 | . . . . . 6 |
11 | 10 | 3impb 1134 | . . . . 5 |
12 | 11 | adantl 271 | . . . 4 |
13 | 2, 12 | supval2ti 6408 | . . 3 |
14 | 3simpc 937 | . . . . 5 | |
15 | 14 | adantl 271 | . . . 4 |
16 | simpr1 944 | . . . . 5 | |
17 | 2, 12 | supeuti 6407 | . . . . 5 |
18 | 9 | riota2 5510 | . . . . 5 |
19 | 16, 17, 18 | syl2anc 403 | . . . 4 |
20 | 15, 19 | mpbid 145 | . . 3 |
21 | 13, 20 | eqtrd 2113 | . 2 |
22 | 21 | ex 113 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 w3a 919 wceq 1284 wcel 1433 wral 2348 wrex 2349 wreu 2350 class class class wbr 3785 crio 5487 csup 6395 |
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-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
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-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-iota 4887 df-riota 5488 df-sup 6397 |
This theorem is referenced by: eqsuptid 6410 eqinfti 6433 maxabs 10095 bezoutlemsup 10398 |
Copyright terms: Public domain | W3C validator |