Theorem supxrbnd 11303

Description: The supremum of a bounded-above nonempty set of reals is real. (Contributed by NM, 19-Jan-2006.)

Assertion

Ref	Expression
supxrbnd	|- ( ( A C_ RR /\ A =/= (/) /\ sup ( A , RR* , < ) < +oo ) -> sup ( A , RR* , < ) e. RR )

Proof of Theorem supxrbnd

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ressxr 9439	. . . . 5 |- RR C_ RR*
2		sstr 3376	. . . . 5 |- ( ( A C_ RR /\ RR C_ RR* ) -> A C_ RR* )
3	1, 2	mpan2 671	. . . 4 |- ( A C_ RR -> A C_ RR* )
4		supxrcl 11289	. . . . . . 7 |- ( A C_ RR* -> sup ( A , RR* , < ) e. RR* )
5		pnfxr 11104	. . . . . . . . . 10 |- +oo e. RR*
6		xrltne 11149	. . . . . . . . . 10 |- ( ( sup ( A , RR* , < ) e. RR* /\ +oo e. RR* /\ sup ( A , RR* , < ) < +oo ) -> +oo =/= sup ( A , RR* , < ) )
7	5, 6	mp3an2 1302	. . . . . . . . 9 |- ( ( sup ( A , RR* , < ) e. RR* /\ sup ( A , RR* , < ) < +oo ) -> +oo =/= sup ( A , RR* , < ) )
8	7	necomd 2707	. . . . . . . 8 |- ( ( sup ( A , RR* , < ) e. RR* /\ sup ( A , RR* , < ) < +oo ) -> sup ( A , RR* , < ) =/= +oo )
9	8	ex 434	. . . . . . 7 |- ( sup ( A , RR* , < ) e. RR* -> ( sup ( A , RR* , < ) < +oo -> sup ( A , RR* , < ) =/= +oo ) )
10	4, 9	syl 16	. . . . . 6 |- ( A C_ RR* -> ( sup ( A , RR* , < ) < +oo -> sup ( A , RR* , < ) =/= +oo ) )
11		supxrunb2 11295	. . . . . . . . 9 |- ( A C_ RR* -> ( A. x e. RR E. y e. A x < y <-> sup ( A , RR* , < ) = +oo ) )
12		ssel2 3363	. . . . . . . . . . . . . 14 |- ( ( A C_ RR* /\ y e. A ) -> y e. RR* )
13	12	adantlr 714	. . . . . . . . . . . . 13 |- ( ( ( A C_ RR* /\ x e. RR ) /\ y e. A ) -> y e. RR* )
14		rexr 9441	. . . . . . . . . . . . . 14 |- ( x e. RR -> x e. RR* )
15	14	ad2antlr 726	. . . . . . . . . . . . 13 |- ( ( ( A C_ RR* /\ x e. RR ) /\ y e. A ) -> x e. RR* )
16		xrlenlt 9454	. . . . . . . . . . . . . 14 |- ( ( y e. RR* /\ x e. RR* ) -> ( y <_ x <-> -. x < y ) )
17	16	con2bid 329	. . . . . . . . . . . . 13 |- ( ( y e. RR* /\ x e. RR* ) -> ( x < y <-> -. y <_ x ) )
18	13, 15, 17	syl2anc 661	. . . . . . . . . . . 12 |- ( ( ( A C_ RR* /\ x e. RR ) /\ y e. A ) -> ( x < y <-> -. y <_ x ) )
19	18	rexbidva 2744	. . . . . . . . . . 11 |- ( ( A C_ RR* /\ x e. RR ) -> ( E. y e. A x < y <-> E. y e. A -. y <_ x ) )
20		rexnal 2738	. . . . . . . . . . 11 |- ( E. y e. A -. y <_ x <-> -. A. y e. A y <_ x )
21	19, 20	syl6bb 261	. . . . . . . . . 10 |- ( ( A C_ RR* /\ x e. RR ) -> ( E. y e. A x < y <-> -. A. y e. A y <_ x ) )
22	21	ralbidva 2743	. . . . . . . . 9 |- ( A C_ RR* -> ( A. x e. RR E. y e. A x < y <-> A. x e. RR -. A. y e. A y <_ x ) )
23	11, 22	bitr3d 255	. . . . . . . 8 |- ( A C_ RR* -> ( sup ( A , RR* , < ) = +oo <-> A. x e. RR -. A. y e. A y <_ x ) )
24		ralnex 2737	. . . . . . . 8 |- ( A. x e. RR -. A. y e. A y <_ x <-> -. E. x e. RR A. y e. A y <_ x )
25	23, 24	syl6bb 261	. . . . . . 7 |- ( A C_ RR* -> ( sup ( A , RR* , < ) = +oo <-> -. E. x e. RR A. y e. A y <_ x ) )
26	25	necon2abid 2680	. . . . . 6 |- ( A C_ RR* -> ( E. x e. RR A. y e. A y <_ x <-> sup ( A , RR* , < ) =/= +oo ) )
27	10, 26	sylibrd 234	. . . . 5 |- ( A C_ RR* -> ( sup ( A , RR* , < ) < +oo -> E. x e. RR A. y e. A y <_ x ) )
28	27	imp 429	. . . 4 |- ( ( A C_ RR* /\ sup ( A , RR* , < ) < +oo ) -> E. x e. RR A. y e. A y <_ x )
29	3, 28	sylan 471	. . 3 |- ( ( A C_ RR /\ sup ( A , RR* , < ) < +oo ) -> E. x e. RR A. y e. A y <_ x )
30	29	3adant2 1007	. 2 |- ( ( A C_ RR /\ A =/= (/) /\ sup ( A , RR* , < ) < +oo ) -> E. x e. RR A. y e. A y <_ x )
31		supxrre 11302	. . 3 |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> sup ( A , RR* , < ) = sup ( A , RR , < ) )
32		suprcl 10302	. . 3 |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> sup ( A , RR , < ) e. RR )
33	31, 32	eqeltrd 2517	. 2 |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> sup ( A , RR* , < ) e. RR )
34	30, 33	syld3an3 1263	1 |- ( ( A C_ RR /\ A =/= (/) /\ sup ( A , RR* , < ) < +oo ) -> sup ( A , RR* , < ) e. RR )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   =/= wne 2618   A.wral 2727   E.wrex 2728   C_ wss 3340   (/)c0 3649   class class class wbr 4304   supcsup 7702   RRcr 9293   +oocpnf 9427   RR*cxr 9429   < clt 9430   <_ cle 9431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-id 4648  df-po 4653  df-so 4654  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-er 7113  df-en 7323  df-dom 7324  df-sdom 7325  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610

This theorem is referenced by:  supxrgtmnf  11304  ovolunlem1  20992  uniioombllem1  21073