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Mirrors > Home > MPE Home > Th. List > suprcl | Structured version Visualization version Unicode version |
Description: Closure of supremum of a nonempty bounded set of reals. (Contributed by NM, 12-Oct-2004.) |
Ref | Expression |
---|---|
suprcl |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltso 9711 |
. . 3
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2 | 1 | a1i 11 |
. 2
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3 | sup3 10563 |
. 2
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4 | 2, 3 | supcl 7969 |
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 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 ax-resscn 9593 ax-1cn 9594 ax-icn 9595 ax-addcl 9596 ax-addrcl 9597 ax-mulcl 9598 ax-mulrcl 9599 ax-mulcom 9600 ax-addass 9601 ax-mulass 9602 ax-distr 9603 ax-i2m1 9604 ax-1ne0 9605 ax-1rid 9606 ax-rnegex 9607 ax-rrecex 9608 ax-cnre 9609 ax-pre-lttri 9610 ax-pre-lttrn 9611 ax-pre-ltadd 9612 ax-pre-mulgt0 9613 ax-pre-sup 9614 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 985 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-nel 2624 df-ral 2741 df-rex 2742 df-reu 2743 df-rmo 2744 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-op 3974 df-uni 4198 df-br 4402 df-opab 4461 df-mpt 4462 df-id 4748 df-po 4754 df-so 4755 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-riota 6250 df-ov 6291 df-oprab 6292 df-mpt2 6293 df-er 7360 df-en 7567 df-dom 7568 df-sdom 7569 df-sup 7953 df-pnf 9674 df-mnf 9675 df-xr 9676 df-ltxr 9677 df-le 9678 df-sub 9859 df-neg 9860 |
This theorem is referenced by: suprub 10567 suprleub 10570 supaddc 10571 supadd 10572 supmul1 10573 supmullem1 10574 supmullem2 10575 supmul 10576 suprclii 10578 infmrclOLD 10590 suprzcl 11012 supminf 11247 supminfOLD 11248 rpnnen1lem4 11290 supxrre 11610 supxrbnd 11611 supicc 11777 flval3 12047 sqrlem4 13302 climsup 13726 supcvg 13907 mertenslem1 13933 ruclem12 14286 prmreclem6 14858 icccmplem2 21834 icccmplem3 21835 reconnlem2 21838 evth 21980 ivthlem2 22396 ivthlem3 22397 ioombl1lem4 22507 mbfsup 22613 mbflimsup 22616 mbflimsupOLD 22617 itg2monolem1 22701 itg2mono 22704 itg2cnlem1 22712 c1liplem1 22941 nmcexi 27672 rge0scvg 28748 ismblfin 31974 itg2addnclem2 31987 ftc1anclem7 32016 ftc1anc 32018 suprcld 36597 ubelsupr 37335 suprnmpt 37433 upbdrech 37517 suprltrp 37545 supsubc 37570 |
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