Theorem ressioosup 39782

Description: If the supremum does not belong to a set of reals, the set is a subset of the unbounded below, right-open interval, with upper bound equal to the supremum. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

Hypotheses

Ref	Expression
ressioosup.a	|- ( ph -> A C_ RR )
ressioosup.s	|- S = sup ( A , RR* , < )
ressioosup.n	|- ( ph -> -. S e. A )
ressioosup.i	|- I = ( -oo (,) S )

Assertion

Ref	Expression
ressioosup	|- ( ph -> A C_ I )

Proof of Theorem ressioosup

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mnfxr 10096	. . . . . 6 |- -oo e. RR*
2	1	a1i 11	. . . . 5 |- ( ( ph /\ x e. A ) -> -oo e. RR* )
3		ressioosup.s	. . . . . 6 |- S = sup ( A , RR* , < )
4		ressioosup.a	. . . . . . . . 9 |- ( ph -> A C_ RR )
5		ressxr 10083	. . . . . . . . . 10 |- RR C_ RR*
6	5	a1i 11	. . . . . . . . 9 |- ( ph -> RR C_ RR* )
7	4, 6	sstrd 3613	. . . . . . . 8 |- ( ph -> A C_ RR* )
8	7	adantr 481	. . . . . . 7 |- ( ( ph /\ x e. A ) -> A C_ RR* )
9	8	supxrcld 39290	. . . . . 6 |- ( ( ph /\ x e. A ) -> sup ( A , RR* , < ) e. RR* )
10	3, 9	syl5eqel 2705	. . . . 5 |- ( ( ph /\ x e. A ) -> S e. RR* )
11	4	adantr 481	. . . . . 6 |- ( ( ph /\ x e. A ) -> A C_ RR )
12		simpr 477	. . . . . 6 |- ( ( ph /\ x e. A ) -> x e. A )
13	11, 12	sseldd 3604	. . . . 5 |- ( ( ph /\ x e. A ) -> x e. RR )
14	13	mnfltd 11958	. . . . 5 |- ( ( ph /\ x e. A ) -> -oo < x )
15	7	sselda 3603	. . . . . 6 |- ( ( ph /\ x e. A ) -> x e. RR* )
16		supxrub 12154	. . . . . . . 8 |- ( ( A C_ RR* /\ x e. A ) -> x <_ sup ( A , RR* , < ) )
17	8, 12, 16	syl2anc 693	. . . . . . 7 |- ( ( ph /\ x e. A ) -> x <_ sup ( A , RR* , < ) )
18	3	a1i 11	. . . . . . . 8 |- ( ( ph /\ x e. A ) -> S = sup ( A , RR* , < ) )
19	18	eqcomd 2628	. . . . . . 7 |- ( ( ph /\ x e. A ) -> sup ( A , RR* , < ) = S )
20	17, 19	breqtrd 4679	. . . . . 6 |- ( ( ph /\ x e. A ) -> x <_ S )
21		id 22	. . . . . . . . . . . 12 |- ( x = S -> x = S )
22	21	eqcomd 2628	. . . . . . . . . . 11 |- ( x = S -> S = x )
23	22	adantl 482	. . . . . . . . . 10 |- ( ( x e. A /\ x = S ) -> S = x )
24		simpl 473	. . . . . . . . . 10 |- ( ( x e. A /\ x = S ) -> x e. A )
25	23, 24	eqeltrd 2701	. . . . . . . . 9 |- ( ( x e. A /\ x = S ) -> S e. A )
26	25	adantll 750	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ x = S ) -> S e. A )
27		ressioosup.n	. . . . . . . . 9 |- ( ph -> -. S e. A )
28	27	ad2antrr 762	. . . . . . . 8 |- ( ( ( ph /\ x e. A ) /\ x = S ) -> -. S e. A )
29	26, 28	pm2.65da 600	. . . . . . 7 |- ( ( ph /\ x e. A ) -> -. x = S )
30	29	neqned 2801	. . . . . 6 |- ( ( ph /\ x e. A ) -> x =/= S )
31	15, 10, 20, 30	xrleneltd 39539	. . . . 5 |- ( ( ph /\ x e. A ) -> x < S )
32	2, 10, 13, 14, 31	eliood 39720	. . . 4 |- ( ( ph /\ x e. A ) -> x e. ( -oo (,) S ) )
33		ressioosup.i	. . . 4 |- I = ( -oo (,) S )
34	32, 33	syl6eleqr 2712	. . 3 |- ( ( ph /\ x e. A ) -> x e. I )
35	34	ralrimiva 2966	. 2 |- ( ph -> A. x e. A x e. I )
36		dfss3 3592	. 2 |- ( A C_ I <-> A. x e. A x e. I )
37	35, 36	sylibr 224	1 |- ( ph -> A C_ I )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   class class class wbr 4653  (class class class)co 6650   supcsup 8346   RRcr 9935   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   (,)cioo 12175

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-ioo 12179

This theorem is referenced by:  pimdecfgtioo  40927  pimincfltioo  40928