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Mirrors > Home > ILE Home > Th. List > suprnubex | Unicode version |
Description: An upper bound is not less than the supremum of a nonempty bounded set of reals. (Contributed by Jim Kingdon, 19-Jan-2022.) |
Ref | Expression |
---|---|
suprubex.ex |
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suprubex.ss |
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suprlubex.b |
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Ref | Expression |
---|---|
suprnubex |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suprubex.ex |
. . . 4
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2 | suprubex.ss |
. . . 4
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3 | suprlubex.b |
. . . 4
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4 | 1, 2, 3 | suprlubex 8568 |
. . 3
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5 | 4 | notbid 633 |
. 2
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6 | ralnex 2385 |
. 2
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7 | 5, 6 | syl6bbr 197 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 584 ax-in2 585 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-13 1459 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 ax-un 4293 ax-setind 4390 ax-cnex 7586 ax-resscn 7587 ax-pre-ltirr 7607 ax-pre-ltwlin 7608 ax-pre-lttrn 7609 ax-pre-apti 7610 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-fal 1305 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ne 2268 df-nel 2363 df-ral 2380 df-rex 2381 df-reu 2382 df-rmo 2383 df-rab 2384 df-v 2643 df-sbc 2863 df-dif 3023 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-po 4156 df-iso 4157 df-xp 4483 df-iota 5024 df-riota 5662 df-sup 6786 df-pnf 7674 df-mnf 7675 df-ltxr 7677 |
This theorem is referenced by: suprleubex 8570 |
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