Theorem supxrbnd 11491

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 9587	. . . . 5 |- RR C_ RR*
2		sstr 3449	. . . . 5 |- ( ( A C_ RR /\ RR C_ RR* ) -> A C_ RR* )
3	1, 2	mpan2 669	. . . 4 |- ( A C_ RR -> A C_ RR* )
4		supxrcl 11477	. . . . . . 7 |- ( A C_ RR* -> sup ( A , RR* , < ) e. RR* )
5		pnfxr 11292	. . . . . . . . . 10 |- +oo e. RR*
6		xrltne 11337	. . . . . . . . . 10 |- ( ( sup ( A , RR* , < ) e. RR* /\ +oo e. RR* /\ sup ( A , RR* , < ) < +oo ) -> +oo =/= sup ( A , RR* , < ) )
7	5, 6	mp3an2 1314	. . . . . . . . 9 |- ( ( sup ( A , RR* , < ) e. RR* /\ sup ( A , RR* , < ) < +oo ) -> +oo =/= sup ( A , RR* , < ) )
8	7	necomd 2674	. . . . . . . 8 |- ( ( sup ( A , RR* , < ) e. RR* /\ sup ( A , RR* , < ) < +oo ) -> sup ( A , RR* , < ) =/= +oo )
9	8	ex 432	. . . . . . 7 |- ( sup ( A , RR* , < ) e. RR* -> ( sup ( A , RR* , < ) < +oo -> sup ( A , RR* , < ) =/= +oo ) )
10	4, 9	syl 17	. . . . . 6 |- ( A C_ RR* -> ( sup ( A , RR* , < ) < +oo -> sup ( A , RR* , < ) =/= +oo ) )
11		supxrunb2 11483	. . . . . . . . 9 |- ( A C_ RR* -> ( A. x e. RR E. y e. A x < y <-> sup ( A , RR* , < ) = +oo ) )
12		ssel2 3436	. . . . . . . . . . . . . 14 |- ( ( A C_ RR* /\ y e. A ) -> y e. RR* )
13	12	adantlr 713	. . . . . . . . . . . . 13 |- ( ( ( A C_ RR* /\ x e. RR ) /\ y e. A ) -> y e. RR* )
14		rexr 9589	. . . . . . . . . . . . . 14 |- ( x e. RR -> x e. RR* )
15	14	ad2antlr 725	. . . . . . . . . . . . 13 |- ( ( ( A C_ RR* /\ x e. RR ) /\ y e. A ) -> x e. RR* )
16		xrlenlt 9602	. . . . . . . . . . . . . 14 |- ( ( y e. RR* /\ x e. RR* ) -> ( y <_ x <-> -. x < y ) )
17	16	con2bid 327	. . . . . . . . . . . . 13 |- ( ( y e. RR* /\ x e. RR* ) -> ( x < y <-> -. y <_ x ) )
18	13, 15, 17	syl2anc 659	. . . . . . . . . . . 12 |- ( ( ( A C_ RR* /\ x e. RR ) /\ y e. A ) -> ( x < y <-> -. y <_ x ) )
19	18	rexbidva 2914	. . . . . . . . . . 11 |- ( ( A C_ RR* /\ x e. RR ) -> ( E. y e. A x < y <-> E. y e. A -. y <_ x ) )
20		rexnal 2851	. . . . . . . . . . 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 2839	. . . . . . . . 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 2849	. . . . . . . 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 2657	. . . . . 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 427	. . . 4 |- ( ( A C_ RR* /\ sup ( A , RR* , < ) < +oo ) -> E. x e. RR A. y e. A y <_ x )
29	3, 28	sylan 469	. . 3 |- ( ( A C_ RR /\ sup ( A , RR* , < ) < +oo ) -> E. x e. RR A. y e. A y <_ x )
30	29	3adant2 1016	. 2 |- ( ( A C_ RR /\ A =/= (/) /\ sup ( A , RR* , < ) < +oo ) -> E. x e. RR A. y e. A y <_ x )
31		supxrre 11490	. . 3 |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> sup ( A , RR* , < ) = sup ( A , RR , < ) )
32		suprcl 10463	. . 3 |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> sup ( A , RR , < ) e. RR )
33	31, 32	eqeltrd 2490	. 2 |- ( ( A C_ RR /\ A =/= (/) /\ E. x e. RR A. y e. A y <_ x ) -> sup ( A , RR* , < ) e. RR )
34	30, 33	syld3an3 1275	1 |- ( ( A C_ RR /\ A =/= (/) /\ sup ( A , RR* , < ) < +oo ) -> sup ( A , RR* , < ) e. RR )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 367   /\ w3a 974   = wceq 1405   e. wcel 1842   =/= wne 2598   A.wral 2753   E.wrex 2754   C_ wss 3413   (/)c0 3737   class class class wbr 4394   supcsup 7854   RRcr 9441   +oocpnf 9575   RR*cxr 9577   < clt 9578   <_ cle 9579

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518  ax-pre-mulgt0 9519  ax-pre-sup 9520

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-sup 7855  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764

This theorem is referenced by:  supxrgtmnf  11492  ovolunlem1  22092  uniioombllem1  22174