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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 |
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supmaxti.2 |
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supmaxti.3 |
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supmaxti.4 |
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Ref | Expression |
---|---|
supmaxti |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | supmaxti.ti |
. 2
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2 | supmaxti.2 |
. 2
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3 | supmaxti.4 |
. 2
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4 | supmaxti.3 |
. . 3
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5 | simprr 531 |
. . 3
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6 | breq2 4007 |
. . . 4
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7 | 6 | rspcev 2841 |
. . 3
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8 | 4, 5, 7 | syl2an2r 595 |
. 2
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9 | 1, 2, 3, 8 | eqsuptid 6995 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-un 3133 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-iota 5178 df-riota 5830 df-sup 6982 |
This theorem is referenced by: supsnti 7003 sup3exmid 8912 maxleim 11209 xrmaxleim 11247 supfz 14700 |
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