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Mirrors > Home > ILE Home > Th. List > supelti | Unicode version |
Description: Supremum membership in a set. (Contributed by Jim Kingdon, 16-Jan-2022.) |
Ref | Expression |
---|---|
supelti.ti | |
supelti.ex | |
supelti.ss |
Ref | Expression |
---|---|
supelti |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supelti.ti | . . . . 5 | |
2 | supelti.ss | . . . . . 6 | |
3 | supelti.ex | . . . . . 6 | |
4 | ssrexv 3059 | . . . . . 6 | |
5 | 2, 3, 4 | sylc 61 | . . . . 5 |
6 | 1, 5 | supclti 6411 | . . . 4 |
7 | elisset 2613 | . . . 4 | |
8 | 6, 7 | syl 14 | . . 3 |
9 | eqcom 2083 | . . . 4 | |
10 | 9 | exbii 1536 | . . 3 |
11 | 8, 10 | sylib 120 | . 2 |
12 | simpr 108 | . . 3 | |
13 | 1, 5 | supval2ti 6408 | . . . . . . . 8 |
14 | 13 | eqeq1d 2089 | . . . . . . 7 |
15 | 14 | biimpa 290 | . . . . . 6 |
16 | 1, 5 | supeuti 6407 | . . . . . . . 8 |
17 | riota1 5506 | . . . . . . . 8 | |
18 | 16, 17 | syl 14 | . . . . . . 7 |
19 | 18 | adantr 270 | . . . . . 6 |
20 | 15, 19 | mpbird 165 | . . . . 5 |
21 | 20 | simpld 110 | . . . 4 |
22 | 2, 3, 16 | jca32 303 | . . . . 5 |
23 | 20 | simprd 112 | . . . . 5 |
24 | reupick 3248 | . . . . 5 | |
25 | 22, 23, 24 | syl2an2r 559 | . . . 4 |
26 | 21, 25 | mpbird 165 | . . 3 |
27 | 12, 26 | eqeltrd 2155 | . 2 |
28 | 11, 27 | exlimddv 1819 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 102 wb 103 wceq 1284 wex 1421 wcel 1433 wral 2348 wrex 2349 wreu 2350 wss 2973 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-in 2979 df-ss 2986 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: zsupcl 10343 |
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