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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 521 | . . 3 | |
6 | breq2 3933 | . . . 4 | |
7 | 6 | rspcev 2789 | . . 3 |
8 | 4, 5, 7 | syl2an2r 584 | . 2 |
9 | 1, 2, 3, 8 | eqsuptid 6884 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wrex 2417 class class class wbr 3929 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: supsnti 6892 sup3exmid 8715 maxleim 10977 xrmaxleim 11013 supfz 13237 |
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