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Mirrors > Home > ILE Home > Th. List > ublbneg | Unicode version |
Description: The image under negation of a bounded-above set of reals is bounded below. For a theorem which is similar but also adds that the bounds need to be the tightest possible, see supinfneg 9390. (Contributed by Paul Chapman, 21-Mar-2011.) |
Ref | Expression |
---|---|
ublbneg |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq1 3932 | . . . . 5 | |
2 | 1 | cbvralv 2654 | . . . 4 |
3 | 2 | rexbii 2442 | . . 3 |
4 | breq2 3933 | . . . . 5 | |
5 | 4 | ralbidv 2437 | . . . 4 |
6 | 5 | cbvrexv 2655 | . . 3 |
7 | 3, 6 | bitri 183 | . 2 |
8 | renegcl 8023 | . . . 4 | |
9 | elrabi 2837 | . . . . . . . . 9 | |
10 | negeq 7955 | . . . . . . . . . . . 12 | |
11 | 10 | eleq1d 2208 | . . . . . . . . . . 11 |
12 | 11 | elrab3 2841 | . . . . . . . . . 10 |
13 | 12 | biimpd 143 | . . . . . . . . 9 |
14 | 9, 13 | mpcom 36 | . . . . . . . 8 |
15 | breq1 3932 | . . . . . . . . 9 | |
16 | 15 | rspcv 2785 | . . . . . . . 8 |
17 | 14, 16 | syl 14 | . . . . . . 7 |
18 | 17 | adantl 275 | . . . . . 6 |
19 | lenegcon1 8228 | . . . . . . 7 | |
20 | 9, 19 | sylan2 284 | . . . . . 6 |
21 | 18, 20 | sylibrd 168 | . . . . 5 |
22 | 21 | ralrimdva 2512 | . . . 4 |
23 | breq1 3932 | . . . . . 6 | |
24 | 23 | ralbidv 2437 | . . . . 5 |
25 | 24 | rspcev 2789 | . . . 4 |
26 | 8, 22, 25 | syl6an 1410 | . . 3 |
27 | 26 | rexlimiv 2543 | . 2 |
28 | 7, 27 | sylbir 134 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 wrex 2417 crab 2420 class class class wbr 3929 cr 7619 cle 7801 cneg 7934 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-addcom 7720 ax-addass 7722 ax-distr 7724 ax-i2m1 7725 ax-0id 7728 ax-rnegex 7729 ax-cnre 7731 ax-pre-ltadd 7736 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 |
This theorem is referenced by: (None) |
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