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Mirrors > Home > MPE Home > Th. List > xrsupss | Structured version Visualization version Unicode version |
Description: Any subset of extended reals has a supremum. (Contributed by NM, 25-Oct-2005.) |
Ref | Expression |
---|---|
xrsupss |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xrsupsslem 11621 |
. 2
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2 | ssdifss 3576 |
. . . 4
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3 | ssxr 9729 |
. . . . . 6
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4 | df-3or 992 |
. . . . . . 7
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5 | neldifsn 4112 |
. . . . . . . 8
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6 | 5 | biorfi 413 |
. . . . . . 7
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7 | 4, 6 | bitr4i 260 |
. . . . . 6
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8 | 3, 7 | sylib 201 |
. . . . 5
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9 | xrsupsslem 11621 |
. . . . 5
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10 | 8, 9 | mpdan 679 |
. . . 4
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11 | 2, 10 | syl 17 |
. . 3
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12 | xrsupexmnf 11619 |
. . . 4
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13 | snssi 4129 |
. . . . . 6
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14 | undif 3860 |
. . . . . . . 8
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15 | uncom 3590 |
. . . . . . . . 9
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16 | 15 | eqeq1i 2467 |
. . . . . . . 8
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17 | 14, 16 | bitri 257 |
. . . . . . 7
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18 | raleq 2999 |
. . . . . . . 8
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19 | rexeq 3000 |
. . . . . . . . . 10
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20 | 19 | imbi2d 322 |
. . . . . . . . 9
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21 | 20 | ralbidv 2839 |
. . . . . . . 8
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22 | 18, 21 | anbi12d 722 |
. . . . . . 7
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23 | 17, 22 | sylbi 200 |
. . . . . 6
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24 | 13, 23 | syl 17 |
. . . . 5
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25 | 24 | rexbidv 2913 |
. . . 4
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26 | 12, 25 | syl5ib 227 |
. . 3
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27 | 11, 26 | mpan9 476 |
. 2
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28 | ssxr 9729 |
. . 3
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
29 | df-3or 992 |
. . 3
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30 | 28, 29 | sylib 201 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
31 | 1, 27, 30 | mpjaodan 800 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1680 ax-4 1693 ax-5 1769 ax-6 1816 ax-7 1862 ax-8 1900 ax-9 1907 ax-10 1926 ax-11 1931 ax-12 1944 ax-13 2102 ax-ext 2442 ax-sep 4539 ax-nul 4548 ax-pow 4595 ax-pr 4653 ax-un 6610 ax-cnex 9621 ax-resscn 9622 ax-1cn 9623 ax-icn 9624 ax-addcl 9625 ax-addrcl 9626 ax-mulcl 9627 ax-mulrcl 9628 ax-mulcom 9629 ax-addass 9630 ax-mulass 9631 ax-distr 9632 ax-i2m1 9633 ax-1ne0 9634 ax-1rid 9635 ax-rnegex 9636 ax-rrecex 9637 ax-cnre 9638 ax-pre-lttri 9639 ax-pre-lttrn 9640 ax-pre-ltadd 9641 ax-pre-mulgt0 9642 ax-pre-sup 9643 |
This theorem depends on definitions: df-bi 190 df-or 376 df-an 377 df-3or 992 df-3an 993 df-tru 1458 df-ex 1675 df-nf 1679 df-sb 1809 df-eu 2314 df-mo 2315 df-clab 2449 df-cleq 2455 df-clel 2458 df-nfc 2592 df-ne 2635 df-nel 2636 df-ral 2754 df-rex 2755 df-reu 2756 df-rab 2758 df-v 3059 df-sbc 3280 df-csb 3376 df-dif 3419 df-un 3421 df-in 3423 df-ss 3430 df-nul 3744 df-if 3894 df-pw 3965 df-sn 3981 df-pr 3983 df-op 3987 df-uni 4213 df-br 4417 df-opab 4476 df-mpt 4477 df-id 4768 df-po 4774 df-so 4775 df-xp 4859 df-rel 4860 df-cnv 4861 df-co 4862 df-dm 4863 df-rn 4864 df-res 4865 df-ima 4866 df-iota 5565 df-fun 5603 df-fn 5604 df-f 5605 df-f1 5606 df-fo 5607 df-f1o 5608 df-fv 5609 df-riota 6277 df-ov 6318 df-oprab 6319 df-mpt2 6320 df-er 7389 df-en 7596 df-dom 7597 df-sdom 7598 df-pnf 9703 df-mnf 9704 df-xr 9705 df-ltxr 9706 df-le 9707 df-sub 9888 df-neg 9889 |
This theorem is referenced by: supxrcl 11629 supxrun 11630 supxrunb1 11634 supxrunb2 11635 supxrub 11639 supxrlub 11640 xrsupssd 28388 xrsclat 28491 itg2addnclem 32038 |
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