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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 527 | . . 3 | |
6 | breq2 3993 | . . . 4 | |
7 | 6 | rspcev 2834 | . . 3 |
8 | 4, 5, 7 | syl2an2r 590 | . 2 |
9 | 1, 2, 3, 8 | eqsuptid 6974 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wceq 1348 wcel 2141 wrex 2449 class class class wbr 3989 csup 6959 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-un 3125 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-br 3990 df-iota 5160 df-riota 5809 df-sup 6961 |
This theorem is referenced by: supsnti 6982 sup3exmid 8873 maxleim 11169 xrmaxleim 11207 supfz 14100 |
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