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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 469 | . . . 4 |
3 | breq1 3979 | . . . . . . . . . 10 | |
4 | 3 | notbid 657 | . . . . . . . . 9 |
5 | 4 | ralbidv 2464 | . . . . . . . 8 |
6 | breq2 3980 | . . . . . . . . . 10 | |
7 | 6 | imbi1d 230 | . . . . . . . . 9 |
8 | 7 | ralbidv 2464 | . . . . . . . 8 |
9 | 5, 8 | anbi12d 465 | . . . . . . 7 |
10 | 9 | rspcev 2825 | . . . . . 6 |
11 | 10 | 3impb 1188 | . . . . 5 |
12 | 11 | adantl 275 | . . . 4 |
13 | 2, 12 | supval2ti 6951 | . . 3 |
14 | 3simpc 985 | . . . . 5 | |
15 | 14 | adantl 275 | . . . 4 |
16 | simpr1 992 | . . . . 5 | |
17 | 2, 12 | supeuti 6950 | . . . . 5 |
18 | 9 | riota2 5814 | . . . . 5 |
19 | 16, 17, 18 | syl2anc 409 | . . . 4 |
20 | 15, 19 | mpbid 146 | . . 3 |
21 | 13, 20 | eqtrd 2197 | . 2 |
22 | 21 | ex 114 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 w3a 967 wceq 1342 wcel 2135 wral 2442 wrex 2443 wreu 2444 class class class wbr 3976 crio 5791 csup 6938 |
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 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-reu 2449 df-rmo 2450 df-rab 2451 df-v 2723 df-sbc 2947 df-un 3115 df-sn 3576 df-pr 3577 df-op 3579 df-uni 3784 df-br 3977 df-iota 5147 df-riota 5792 df-sup 6940 |
This theorem is referenced by: eqsuptid 6953 eqinfti 6976 maxabs 11137 xrmaxif 11178 suprzcl2dc 11873 bezoutlemsup 11927 suplociccex 13144 |
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