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Mirrors > Home > MPE Home > Th. List > supxrbnd | Structured version Visualization version Unicode version |
Description: The supremum of a bounded-above nonempty set of reals is real. (Contributed by NM, 19-Jan-2006.) |
Ref | Expression |
---|---|
supxrbnd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ressxr 9684 |
. . . . 5
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2 | sstr 3440 |
. . . . 5
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3 | 1, 2 | mpan2 677 |
. . . 4
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4 | supxrcl 11600 |
. . . . . . 7
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5 | pnfxr 11412 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() | |
6 | xrltne 11460 |
. . . . . . . . . 10
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7 | 5, 6 | mp3an2 1352 |
. . . . . . . . 9
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8 | 7 | necomd 2679 |
. . . . . . . 8
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9 | 8 | ex 436 |
. . . . . . 7
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10 | 4, 9 | syl 17 |
. . . . . 6
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11 | supxrunb2 11606 |
. . . . . . . . 9
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12 | ssel2 3427 |
. . . . . . . . . . . . . 14
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13 | 12 | adantlr 721 |
. . . . . . . . . . . . 13
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14 | rexr 9686 |
. . . . . . . . . . . . . 14
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15 | 14 | ad2antlr 733 |
. . . . . . . . . . . . 13
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16 | xrlenlt 9699 |
. . . . . . . . . . . . . 14
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17 | 16 | con2bid 331 |
. . . . . . . . . . . . 13
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18 | 13, 15, 17 | syl2anc 667 |
. . . . . . . . . . . 12
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19 | 18 | rexbidva 2898 |
. . . . . . . . . . 11
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20 | rexnal 2836 |
. . . . . . . . . . 11
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
21 | 19, 20 | syl6bb 265 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
22 | 21 | ralbidva 2824 |
. . . . . . . . 9
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23 | 11, 22 | bitr3d 259 |
. . . . . . . 8
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24 | ralnex 2834 |
. . . . . . . 8
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
25 | 23, 24 | syl6bb 265 |
. . . . . . 7
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26 | 25 | necon2abid 2666 |
. . . . . 6
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27 | 10, 26 | sylibrd 238 |
. . . . 5
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28 | 27 | imp 431 |
. . . 4
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29 | 3, 28 | sylan 474 |
. . 3
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30 | 29 | 3adant2 1027 |
. 2
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31 | supxrre 11613 |
. . 3
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32 | suprcl 10569 |
. . 3
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33 | 31, 32 | eqeltrd 2529 |
. 2
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34 | 30, 33 | syld3an3 1313 |
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 1669 ax-4 1682 ax-5 1758 ax-6 1805 ax-7 1851 ax-8 1889 ax-9 1896 ax-10 1915 ax-11 1920 ax-12 1933 ax-13 2091 ax-ext 2431 ax-sep 4525 ax-nul 4534 ax-pow 4581 ax-pr 4639 ax-un 6583 ax-cnex 9595 ax-resscn 9596 ax-1cn 9597 ax-icn 9598 ax-addcl 9599 ax-addrcl 9600 ax-mulcl 9601 ax-mulrcl 9602 ax-mulcom 9603 ax-addass 9604 ax-mulass 9605 ax-distr 9606 ax-i2m1 9607 ax-1ne0 9608 ax-1rid 9609 ax-rnegex 9610 ax-rrecex 9611 ax-cnre 9612 ax-pre-lttri 9613 ax-pre-lttrn 9614 ax-pre-ltadd 9615 ax-pre-mulgt0 9616 ax-pre-sup 9617 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 986 df-3an 987 df-tru 1447 df-ex 1664 df-nf 1668 df-sb 1798 df-eu 2303 df-mo 2304 df-clab 2438 df-cleq 2444 df-clel 2447 df-nfc 2581 df-ne 2624 df-nel 2625 df-ral 2742 df-rex 2743 df-reu 2744 df-rmo 2745 df-rab 2746 df-v 3047 df-sbc 3268 df-csb 3364 df-dif 3407 df-un 3409 df-in 3411 df-ss 3418 df-nul 3732 df-if 3882 df-pw 3953 df-sn 3969 df-pr 3971 df-op 3975 df-uni 4199 df-br 4403 df-opab 4462 df-mpt 4463 df-id 4749 df-po 4755 df-so 4756 df-xp 4840 df-rel 4841 df-cnv 4842 df-co 4843 df-dm 4844 df-rn 4845 df-res 4846 df-ima 4847 df-iota 5546 df-fun 5584 df-fn 5585 df-f 5586 df-f1 5587 df-fo 5588 df-f1o 5589 df-fv 5590 df-riota 6252 df-ov 6293 df-oprab 6294 df-mpt2 6295 df-er 7363 df-en 7570 df-dom 7571 df-sdom 7572 df-sup 7956 df-pnf 9677 df-mnf 9678 df-xr 9679 df-ltxr 9680 df-le 9681 df-sub 9862 df-neg 9863 |
This theorem is referenced by: supxrgtmnf 11615 ovolunlem1 22450 uniioombllem1 22538 |
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