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Mirrors > Home > MPE Home > Th. List > supxrss | Unicode version |
Description: Smaller sets of extended reals have smaller suprema. (Contributed by Mario Carneiro, 1-Apr-2015.) |
Ref | Expression |
---|---|
supxrss |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simplr 732 |
. . . 4
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2 | simpl 444 |
. . . . 5
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3 | 2 | sselda 3308 |
. . . 4
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4 | supxrub 10859 |
. . . 4
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5 | 1, 3, 4 | syl2anc 643 |
. . 3
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6 | 5 | ralrimiva 2749 |
. 2
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7 | sstr 3316 |
. . 3
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8 | supxrcl 10849 |
. . . 4
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9 | 8 | adantl 453 |
. . 3
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10 | supxrleub 10861 |
. . 3
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11 | 7, 9, 10 | syl2anc 643 |
. 2
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12 | 6, 11 | mpbird 224 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: deg1mul3le 19992 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-13 1723 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2385 ax-sep 4290 ax-nul 4298 ax-pow 4337 ax-pr 4363 ax-un 4660 ax-cnex 9002 ax-resscn 9003 ax-1cn 9004 ax-icn 9005 ax-addcl 9006 ax-addrcl 9007 ax-mulcl 9008 ax-mulrcl 9009 ax-mulcom 9010 ax-addass 9011 ax-mulass 9012 ax-distr 9013 ax-i2m1 9014 ax-1ne0 9015 ax-1rid 9016 ax-rnegex 9017 ax-rrecex 9018 ax-cnre 9019 ax-pre-lttri 9020 ax-pre-lttrn 9021 ax-pre-ltadd 9022 ax-pre-mulgt0 9023 ax-pre-sup 9024 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3or 937 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2258 df-mo 2259 df-clab 2391 df-cleq 2397 df-clel 2400 df-nfc 2529 df-ne 2569 df-nel 2570 df-ral 2671 df-rex 2672 df-reu 2673 df-rmo 2674 df-rab 2675 df-v 2918 df-sbc 3122 df-csb 3212 df-dif 3283 df-un 3285 df-in 3287 df-ss 3294 df-nul 3589 df-if 3700 df-pw 3761 df-sn 3780 df-pr 3781 df-op 3783 df-uni 3976 df-br 4173 df-opab 4227 df-mpt 4228 df-id 4458 df-po 4463 df-so 4464 df-xp 4843 df-rel 4844 df-cnv 4845 df-co 4846 df-dm 4847 df-rn 4848 df-res 4849 df-ima 4850 df-iota 5377 df-fun 5415 df-fn 5416 df-f 5417 df-f1 5418 df-fo 5419 df-f1o 5420 df-fv 5421 df-ov 6043 df-oprab 6044 df-mpt2 6045 df-riota 6508 df-er 6864 df-en 7069 df-dom 7070 df-sdom 7071 df-sup 7404 df-pnf 9078 df-mnf 9079 df-xr 9080 df-ltxr 9081 df-le 9082 df-sub 9249 df-neg 9250 |
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