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Mirrors > Home > ILE Home > Th. List > supmaxti | Unicode version |
Description: The greatest element of a set is its supremum. Note that the converse is not true; the supremum might not be an element of the set considered. (Contributed by Jim Kingdon, 24-Nov-2021.) |
Ref | Expression |
---|---|
supmaxti.ti | |
supmaxti.2 | |
supmaxti.3 | |
supmaxti.4 |
Ref | Expression |
---|---|
supmaxti |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supmaxti.ti | . 2 | |
2 | supmaxti.2 | . 2 | |
3 | supmaxti.4 | . 2 | |
4 | supmaxti.3 | . . 3 | |
5 | simprr 522 | . . 3 | |
6 | breq2 3971 | . . . 4 | |
7 | 6 | rspcev 2816 | . . 3 |
8 | 4, 5, 7 | syl2an2r 585 | . 2 |
9 | 1, 2, 3, 8 | eqsuptid 6943 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1335 wcel 2128 wrex 2436 class class class wbr 3967 csup 6928 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-un 3106 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-br 3968 df-iota 5137 df-riota 5782 df-sup 6930 |
This theorem is referenced by: supsnti 6951 sup3exmid 8833 maxleim 11116 xrmaxleim 11152 supfz 13710 |
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