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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 8934 |
. . 3
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5 | 4 | notbid 668 |
. 2
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6 | ralnex 2478 |
. 2
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7 | 5, 6 | bitr4di 198 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-cnex 7927 ax-resscn 7928 ax-pre-ltirr 7948 ax-pre-ltwlin 7949 ax-pre-lttrn 7950 ax-pre-apti 7951 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-br 4019 df-opab 4080 df-po 4311 df-iso 4312 df-xp 4647 df-iota 5193 df-riota 5848 df-sup 7008 df-pnf 8019 df-mnf 8020 df-ltxr 8022 |
This theorem is referenced by: suprleubex 8936 |
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