Theorem supxrgere 39549

Description: If a real number can be approximated from below by members of a set, then it is smaller or equal to the supremum of the set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)

Hypotheses

Ref	Expression
supxrgere.xph	|- F/ x ph
supxrgere.a	|- ( ph -> A C_ RR* )
supxrgere.b	|- ( ph -> B e. RR )
supxrgere.y	|- ( ( ph /\ x e. RR+ ) -> E. y e. A ( B - x ) < y )

Assertion

Ref	Expression
supxrgere	|- ( ph -> B <_ sup ( A , RR* , < ) )

Distinct variable groups:   x, A, y   x, B, y   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem supxrgere

Step	Hyp	Ref	Expression
1		supxrgere.b	. . . . 5 |- ( ph -> B e. RR )
2		rexr 10085	. . . . . 6 |- ( B e. RR -> B e. RR* )
3		pnfxr 10092	. . . . . . 7 |- +oo e. RR*
4	3	a1i 11	. . . . . 6 |- ( B e. RR -> +oo e. RR* )
5		ltpnf 11954	. . . . . 6 |- ( B e. RR -> B < +oo )
6	2, 4, 5	xrltled 39486	. . . . 5 |- ( B e. RR -> B <_ +oo )
7	1, 6	syl 17	. . . 4 |- ( ph -> B <_ +oo )
8	7	adantr 481	. . 3 |- ( ( ph /\ sup ( A , RR* , < ) = +oo ) -> B <_ +oo )
9		id 22	. . . . 5 |- ( sup ( A , RR* , < ) = +oo -> sup ( A , RR* , < ) = +oo )
10	9	eqcomd 2628	. . . 4 |- ( sup ( A , RR* , < ) = +oo -> +oo = sup ( A , RR* , < ) )
11	10	adantl 482	. . 3 |- ( ( ph /\ sup ( A , RR* , < ) = +oo ) -> +oo = sup ( A , RR* , < ) )
12	8, 11	breqtrd 4679	. 2 |- ( ( ph /\ sup ( A , RR* , < ) = +oo ) -> B <_ sup ( A , RR* , < ) )
13		simpl 473	. . 3 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> ph )
14		1rp 11836	. . . . . . . 8 |- 1 e. RR+
15		nfcv 2764	. . . . . . . . . 10 |- F/_ x 1
16		supxrgere.xph	. . . . . . . . . . . 12 |- F/ x ph
17		nfv 1843	. . . . . . . . . . . 12 |- F/ x 1 e. RR+
18	16, 17	nfan 1828	. . . . . . . . . . 11 |- F/ x ( ph /\ 1 e. RR+ )
19		nfv 1843	. . . . . . . . . . 11 |- F/ x E. y e. A ( B - 1 ) < y
20	18, 19	nfim 1825	. . . . . . . . . 10 |- F/ x ( ( ph /\ 1 e. RR+ ) -> E. y e. A ( B - 1 ) < y )
21		eleq1 2689	. . . . . . . . . . . 12 |- ( x = 1 -> ( x e. RR+ <-> 1 e. RR+ ) )
22	21	anbi2d 740	. . . . . . . . . . 11 |- ( x = 1 -> ( ( ph /\ x e. RR+ ) <-> ( ph /\ 1 e. RR+ ) ) )
23		oveq2 6658	. . . . . . . . . . . . 13 |- ( x = 1 -> ( B - x ) = ( B - 1 ) )
24	23	breq1d 4663	. . . . . . . . . . . 12 |- ( x = 1 -> ( ( B - x ) < y <-> ( B - 1 ) < y ) )
25	24	rexbidv 3052	. . . . . . . . . . 11 |- ( x = 1 -> ( E. y e. A ( B - x ) < y <-> E. y e. A ( B - 1 ) < y ) )
26	22, 25	imbi12d 334	. . . . . . . . . 10 |- ( x = 1 -> ( ( ( ph /\ x e. RR+ ) -> E. y e. A ( B - x ) < y ) <-> ( ( ph /\ 1 e. RR+ ) -> E. y e. A ( B - 1 ) < y ) ) )
27		supxrgere.y	. . . . . . . . . 10 |- ( ( ph /\ x e. RR+ ) -> E. y e. A ( B - x ) < y )
28	15, 20, 26, 27	vtoclgf 3264	. . . . . . . . 9 |- ( 1 e. RR+ -> ( ( ph /\ 1 e. RR+ ) -> E. y e. A ( B - 1 ) < y ) )
29	14, 28	ax-mp 5	. . . . . . . 8 |- ( ( ph /\ 1 e. RR+ ) -> E. y e. A ( B - 1 ) < y )
30	14, 29	mpan2 707	. . . . . . 7 |- ( ph -> E. y e. A ( B - 1 ) < y )
31	30	adantr 481	. . . . . 6 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> E. y e. A ( B - 1 ) < y )
32		mnfxr 10096	. . . . . . . . . . 11 |- -oo e. RR*
33	32	a1i 11	. . . . . . . . . 10 |- ( ( ph /\ y e. A /\ ( B - 1 ) < y ) -> -oo e. RR* )
34		supxrgere.a	. . . . . . . . . . . 12 |- ( ph -> A C_ RR* )
35	34	sselda 3603	. . . . . . . . . . 11 |- ( ( ph /\ y e. A ) -> y e. RR* )
36	35	3adant3 1081	. . . . . . . . . 10 |- ( ( ph /\ y e. A /\ ( B - 1 ) < y ) -> y e. RR* )
37		supxrcl 12145	. . . . . . . . . . . 12 |- ( A C_ RR* -> sup ( A , RR* , < ) e. RR* )
38	34, 37	syl 17	. . . . . . . . . . 11 |- ( ph -> sup ( A , RR* , < ) e. RR* )
39	38	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( ph /\ y e. A /\ ( B - 1 ) < y ) -> sup ( A , RR* , < ) e. RR* )
40		peano2rem 10348	. . . . . . . . . . . . . . . 16 |- ( B e. RR -> ( B - 1 ) e. RR )
41	1, 40	syl 17	. . . . . . . . . . . . . . 15 |- ( ph -> ( B - 1 ) e. RR )
42	41	rexrd 10089	. . . . . . . . . . . . . 14 |- ( ph -> ( B - 1 ) e. RR* )
43	42	adantr 481	. . . . . . . . . . . . 13 |- ( ( ph /\ -. -oo < y ) -> ( B - 1 ) e. RR* )
44	43	3ad2antl1 1223	. . . . . . . . . . . 12 |- ( ( ( ph /\ y e. A /\ ( B - 1 ) < y ) /\ -. -oo < y ) -> ( B - 1 ) e. RR* )
45	36	adantr 481	. . . . . . . . . . . 12 |- ( ( ( ph /\ y e. A /\ ( B - 1 ) < y ) /\ -. -oo < y ) -> y e. RR* )
46	32	a1i 11	. . . . . . . . . . . 12 |- ( ( ( ph /\ y e. A /\ ( B - 1 ) < y ) /\ -. -oo < y ) -> -oo e. RR* )
47		simpl3 1066	. . . . . . . . . . . 12 |- ( ( ( ph /\ y e. A /\ ( B - 1 ) < y ) /\ -. -oo < y ) -> ( B - 1 ) < y )
48		simpr 477	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. A ) /\ -. -oo < y ) -> -. -oo < y )
49	35	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ y e. A ) /\ -. -oo < y ) -> y e. RR* )
50	32	a1i 11	. . . . . . . . . . . . . . 15 |- ( ( ( ph /\ y e. A ) /\ -. -oo < y ) -> -oo e. RR* )
51		xrlenlt 10103	. . . . . . . . . . . . . . 15 |- ( ( y e. RR* /\ -oo e. RR* ) -> ( y <_ -oo <-> -. -oo < y ) )
52	49, 50, 51	syl2anc 693	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ y e. A ) /\ -. -oo < y ) -> ( y <_ -oo <-> -. -oo < y ) )
53	48, 52	mpbird 247	. . . . . . . . . . . . 13 |- ( ( ( ph /\ y e. A ) /\ -. -oo < y ) -> y <_ -oo )
54	53	3adantl3 1219	. . . . . . . . . . . 12 |- ( ( ( ph /\ y e. A /\ ( B - 1 ) < y ) /\ -. -oo < y ) -> y <_ -oo )
55	44, 45, 46, 47, 54	xrltletrd 11992	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. A /\ ( B - 1 ) < y ) /\ -. -oo < y ) -> ( B - 1 ) < -oo )
56		nltmnf 11963	. . . . . . . . . . . . . 14 |- ( ( B - 1 ) e. RR* -> -. ( B - 1 ) < -oo )
57	42, 56	syl 17	. . . . . . . . . . . . 13 |- ( ph -> -. ( B - 1 ) < -oo )
58	57	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ -. -oo < y ) -> -. ( B - 1 ) < -oo )
59	58	3ad2antl1 1223	. . . . . . . . . . 11 |- ( ( ( ph /\ y e. A /\ ( B - 1 ) < y ) /\ -. -oo < y ) -> -. ( B - 1 ) < -oo )
60	55, 59	condan 835	. . . . . . . . . 10 |- ( ( ph /\ y e. A /\ ( B - 1 ) < y ) -> -oo < y )
61	34	adantr 481	. . . . . . . . . . . 12 |- ( ( ph /\ y e. A ) -> A C_ RR* )
62		simpr 477	. . . . . . . . . . . 12 |- ( ( ph /\ y e. A ) -> y e. A )
63		supxrub 12154	. . . . . . . . . . . 12 |- ( ( A C_ RR* /\ y e. A ) -> y <_ sup ( A , RR* , < ) )
64	61, 62, 63	syl2anc 693	. . . . . . . . . . 11 |- ( ( ph /\ y e. A ) -> y <_ sup ( A , RR* , < ) )
65	64	3adant3 1081	. . . . . . . . . 10 |- ( ( ph /\ y e. A /\ ( B - 1 ) < y ) -> y <_ sup ( A , RR* , < ) )
66	33, 36, 39, 60, 65	xrltletrd 11992	. . . . . . . . 9 |- ( ( ph /\ y e. A /\ ( B - 1 ) < y ) -> -oo < sup ( A , RR* , < ) )
67	66	3exp 1264	. . . . . . . 8 |- ( ph -> ( y e. A -> ( ( B - 1 ) < y -> -oo < sup ( A , RR* , < ) ) ) )
68	67	adantr 481	. . . . . . 7 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> ( y e. A -> ( ( B - 1 ) < y -> -oo < sup ( A , RR* , < ) ) ) )
69	68	rexlimdv 3030	. . . . . 6 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> ( E. y e. A ( B - 1 ) < y -> -oo < sup ( A , RR* , < ) ) )
70	31, 69	mpd 15	. . . . 5 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> -oo < sup ( A , RR* , < ) )
71		simpr 477	. . . . . . 7 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> -. sup ( A , RR* , < ) = +oo )
72		nltpnft 11995	. . . . . . . . 9 |- ( sup ( A , RR* , < ) e. RR* -> ( sup ( A , RR* , < ) = +oo <-> -. sup ( A , RR* , < ) < +oo ) )
73	38, 72	syl 17	. . . . . . . 8 |- ( ph -> ( sup ( A , RR* , < ) = +oo <-> -. sup ( A , RR* , < ) < +oo ) )
74	73	adantr 481	. . . . . . 7 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> ( sup ( A , RR* , < ) = +oo <-> -. sup ( A , RR* , < ) < +oo ) )
75	71, 74	mtbid 314	. . . . . 6 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> -. -. sup ( A , RR* , < ) < +oo )
76	75	notnotrd 128	. . . . 5 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> sup ( A , RR* , < ) < +oo )
77	70, 76	jca 554	. . . 4 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> ( -oo < sup ( A , RR* , < ) /\ sup ( A , RR* , < ) < +oo ) )
78	38	adantr 481	. . . . 5 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> sup ( A , RR* , < ) e. RR* )
79		xrrebnd 11999	. . . . 5 |- ( sup ( A , RR* , < ) e. RR* -> ( sup ( A , RR* , < ) e. RR <-> ( -oo < sup ( A , RR* , < ) /\ sup ( A , RR* , < ) < +oo ) ) )
80	78, 79	syl 17	. . . 4 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> ( sup ( A , RR* , < ) e. RR <-> ( -oo < sup ( A , RR* , < ) /\ sup ( A , RR* , < ) < +oo ) ) )
81	77, 80	mpbird 247	. . 3 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> sup ( A , RR* , < ) e. RR )
82		simpl 473	. . . . 5 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ -. B <_ sup ( A , RR* , < ) ) -> ( ph /\ sup ( A , RR* , < ) e. RR ) )
83		simpr 477	. . . . . 6 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ -. B <_ sup ( A , RR* , < ) ) -> -. B <_ sup ( A , RR* , < ) )
84	82	simprd 479	. . . . . . 7 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ -. B <_ sup ( A , RR* , < ) ) -> sup ( A , RR* , < ) e. RR )
85	1	ad2antrr 762	. . . . . . 7 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ -. B <_ sup ( A , RR* , < ) ) -> B e. RR )
86	84, 85	ltnled 10184	. . . . . 6 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ -. B <_ sup ( A , RR* , < ) ) -> ( sup ( A , RR* , < ) < B <-> -. B <_ sup ( A , RR* , < ) ) )
87	83, 86	mpbird 247	. . . . 5 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ -. B <_ sup ( A , RR* , < ) ) -> sup ( A , RR* , < ) < B )
88		simpll 790	. . . . . . 7 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> ph )
89	1	adantr 481	. . . . . . . . . 10 |- ( ( ph /\ sup ( A , RR* , < ) e. RR ) -> B e. RR )
90		simpr 477	. . . . . . . . . 10 |- ( ( ph /\ sup ( A , RR* , < ) e. RR ) -> sup ( A , RR* , < ) e. RR )
91	89, 90	resubcld 10458	. . . . . . . . 9 |- ( ( ph /\ sup ( A , RR* , < ) e. RR ) -> ( B - sup ( A , RR* , < ) ) e. RR )
92	91	adantr 481	. . . . . . . 8 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> ( B - sup ( A , RR* , < ) ) e. RR )
93		simpr 477	. . . . . . . . 9 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> sup ( A , RR* , < ) < B )
94	90	adantr 481	. . . . . . . . . 10 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> sup ( A , RR* , < ) e. RR )
95	88, 1	syl 17	. . . . . . . . . 10 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> B e. RR )
96	94, 95	posdifd 10614	. . . . . . . . 9 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> ( sup ( A , RR* , < ) < B <-> 0 < ( B - sup ( A , RR* , < ) ) ) )
97	93, 96	mpbid 222	. . . . . . . 8 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> 0 < ( B - sup ( A , RR* , < ) ) )
98	92, 97	elrpd 11869	. . . . . . 7 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> ( B - sup ( A , RR* , < ) ) e. RR+ )
99		ovex 6678	. . . . . . . 8 |- ( B - sup ( A , RR* , < ) ) e. _V
100		nfcv 2764	. . . . . . . . 9 |- F/_ x ( B - sup ( A , RR* , < ) )
101		nfv 1843	. . . . . . . . . . 11 |- F/ x ( B - sup ( A , RR* , < ) ) e. RR+
102	16, 101	nfan 1828	. . . . . . . . . 10 |- F/ x ( ph /\ ( B - sup ( A , RR* , < ) ) e. RR+ )
103		nfv 1843	. . . . . . . . . 10 |- F/ x E. y e. A ( B - ( B - sup ( A , RR* , < ) ) ) < y
104	102, 103	nfim 1825	. . . . . . . . 9 |- F/ x ( ( ph /\ ( B - sup ( A , RR* , < ) ) e. RR+ ) -> E. y e. A ( B - ( B - sup ( A , RR* , < ) ) ) < y )
105		eleq1 2689	. . . . . . . . . . 11 |- ( x = ( B - sup ( A , RR* , < ) ) -> ( x e. RR+ <-> ( B - sup ( A , RR* , < ) ) e. RR+ ) )
106	105	anbi2d 740	. . . . . . . . . 10 |- ( x = ( B - sup ( A , RR* , < ) ) -> ( ( ph /\ x e. RR+ ) <-> ( ph /\ ( B - sup ( A , RR* , < ) ) e. RR+ ) ) )
107		oveq2 6658	. . . . . . . . . . . 12 |- ( x = ( B - sup ( A , RR* , < ) ) -> ( B - x ) = ( B - ( B - sup ( A , RR* , < ) ) ) )
108	107	breq1d 4663	. . . . . . . . . . 11 |- ( x = ( B - sup ( A , RR* , < ) ) -> ( ( B - x ) < y <-> ( B - ( B - sup ( A , RR* , < ) ) ) < y ) )
109	108	rexbidv 3052	. . . . . . . . . 10 |- ( x = ( B - sup ( A , RR* , < ) ) -> ( E. y e. A ( B - x ) < y <-> E. y e. A ( B - ( B - sup ( A , RR* , < ) ) ) < y ) )
110	106, 109	imbi12d 334	. . . . . . . . 9 |- ( x = ( B - sup ( A , RR* , < ) ) -> ( ( ( ph /\ x e. RR+ ) -> E. y e. A ( B - x ) < y ) <-> ( ( ph /\ ( B - sup ( A , RR* , < ) ) e. RR+ ) -> E. y e. A ( B - ( B - sup ( A , RR* , < ) ) ) < y ) ) )
111	100, 104, 110, 27	vtoclgf 3264	. . . . . . . 8 |- ( ( B - sup ( A , RR* , < ) ) e. _V -> ( ( ph /\ ( B - sup ( A , RR* , < ) ) e. RR+ ) -> E. y e. A ( B - ( B - sup ( A , RR* , < ) ) ) < y ) )
112	99, 111	ax-mp 5	. . . . . . 7 |- ( ( ph /\ ( B - sup ( A , RR* , < ) ) e. RR+ ) -> E. y e. A ( B - ( B - sup ( A , RR* , < ) ) ) < y )
113	88, 98, 112	syl2anc 693	. . . . . 6 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> E. y e. A ( B - ( B - sup ( A , RR* , < ) ) ) < y )
114	1	recnd 10068	. . . . . . . . . . . . 13 |- ( ph -> B e. CC )
115	114	ad3antrrr 766	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) /\ ( B - ( B - sup ( A , RR* , < ) ) ) < y ) -> B e. CC )
116	90	recnd 10068	. . . . . . . . . . . . 13 |- ( ( ph /\ sup ( A , RR* , < ) e. RR ) -> sup ( A , RR* , < ) e. CC )
117	116	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) /\ ( B - ( B - sup ( A , RR* , < ) ) ) < y ) -> sup ( A , RR* , < ) e. CC )
118	115, 117	nncand 10397	. . . . . . . . . . 11 |- ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) /\ ( B - ( B - sup ( A , RR* , < ) ) ) < y ) -> ( B - ( B - sup ( A , RR* , < ) ) ) = sup ( A , RR* , < ) )
119	118	eqcomd 2628	. . . . . . . . . 10 |- ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) /\ ( B - ( B - sup ( A , RR* , < ) ) ) < y ) -> sup ( A , RR* , < ) = ( B - ( B - sup ( A , RR* , < ) ) ) )
120		simpr 477	. . . . . . . . . 10 |- ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) /\ ( B - ( B - sup ( A , RR* , < ) ) ) < y ) -> ( B - ( B - sup ( A , RR* , < ) ) ) < y )
121	119, 120	eqbrtrd 4675	. . . . . . . . 9 |- ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) /\ ( B - ( B - sup ( A , RR* , < ) ) ) < y ) -> sup ( A , RR* , < ) < y )
122	121	ex 450	. . . . . . . 8 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> ( ( B - ( B - sup ( A , RR* , < ) ) ) < y -> sup ( A , RR* , < ) < y ) )
123	122	adantr 481	. . . . . . 7 |- ( ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) /\ y e. A ) -> ( ( B - ( B - sup ( A , RR* , < ) ) ) < y -> sup ( A , RR* , < ) < y ) )
124	123	reximdva 3017	. . . . . 6 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> ( E. y e. A ( B - ( B - sup ( A , RR* , < ) ) ) < y -> E. y e. A sup ( A , RR* , < ) < y ) )
125	113, 124	mpd 15	. . . . 5 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ sup ( A , RR* , < ) < B ) -> E. y e. A sup ( A , RR* , < ) < y )
126	82, 87, 125	syl2anc 693	. . . 4 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ -. B <_ sup ( A , RR* , < ) ) -> E. y e. A sup ( A , RR* , < ) < y )
127	61, 37	syl 17	. . . . . . . . 9 |- ( ( ph /\ y e. A ) -> sup ( A , RR* , < ) e. RR* )
128	35, 127	xrlenltd 10104	. . . . . . . 8 |- ( ( ph /\ y e. A ) -> ( y <_ sup ( A , RR* , < ) <-> -. sup ( A , RR* , < ) < y ) )
129	64, 128	mpbid 222	. . . . . . 7 |- ( ( ph /\ y e. A ) -> -. sup ( A , RR* , < ) < y )
130	129	ralrimiva 2966	. . . . . 6 |- ( ph -> A. y e. A -. sup ( A , RR* , < ) < y )
131		ralnex 2992	. . . . . 6 |- ( A. y e. A -. sup ( A , RR* , < ) < y <-> -. E. y e. A sup ( A , RR* , < ) < y )
132	130, 131	sylib 208	. . . . 5 |- ( ph -> -. E. y e. A sup ( A , RR* , < ) < y )
133	132	ad2antrr 762	. . . 4 |- ( ( ( ph /\ sup ( A , RR* , < ) e. RR ) /\ -. B <_ sup ( A , RR* , < ) ) -> -. E. y e. A sup ( A , RR* , < ) < y )
134	126, 133	condan 835	. . 3 |- ( ( ph /\ sup ( A , RR* , < ) e. RR ) -> B <_ sup ( A , RR* , < ) )
135	13, 81, 134	syl2anc 693	. 2 |- ( ( ph /\ -. sup ( A , RR* , < ) = +oo ) -> B <_ sup ( A , RR* , < ) )
136	12, 135	pm2.61dan 832	1 |- ( ph -> B <_ sup ( A , RR* , < ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   F/wnf 1708   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   class class class wbr 4653  (class class class)co 6650   supcsup 8346   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   +oocpnf 10071   -oocmnf 10072   RR*cxr 10073   < clt 10074   <_ cle 10075   - cmin 10266   RR+crp 11832

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-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-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-rp 11833

This theorem is referenced by:  suplesup  39555