Theorem supiccub

Description: The supremum of a bounded set of real numbers is an upper bound. (Contributed by Thierry Arnoux, 20-May-2019)

	Ref	Expression
Hypotheses	supicc.1	|- ( ph -> B e. RR )
	supicc.2	|- ( ph -> C e. RR )
	supicc.3	|- ( ph -> A C_ ( B [,] C ) )
	supicc.4	|- ( ph -> A =/= (/) )
	supiccub.1	|- ( ph -> D e. A )
Assertion	supiccub	|- ( ph -> D <_ sup ( A , RR , < ) )

Proof

Step	Hyp	Ref	Expression
1		supicc.1	|- ( ph -> B e. RR )
2		supicc.2	|- ( ph -> C e. RR )
3		supicc.3	|- ( ph -> A C_ ( B [,] C ) )
4		supicc.4	|- ( ph -> A =/= (/) )
5		supiccub.1	|- ( ph -> D e. A )
6		iccssre	|- ( ( B e. RR /\ C e. RR ) -> ( B [,] C ) C_ RR )
7	1 2 6	syl2anc	|- ( ph -> ( B [,] C ) C_ RR )
8	3 7	sstrd	|- ( ph -> A C_ RR )
9	1	adantr	|- ( ( ph /\ x e. A ) -> B e. RR )
10	9	rexrd	|- ( ( ph /\ x e. A ) -> B e. RR* )
11	2	adantr	|- ( ( ph /\ x e. A ) -> C e. RR )
12	11	rexrd	|- ( ( ph /\ x e. A ) -> C e. RR* )
13	3	sselda	|- ( ( ph /\ x e. A ) -> x e. ( B [,] C ) )
14		iccleub	|- ( ( B e. RR* /\ C e. RR* /\ x e. ( B [,] C ) ) -> x <_ C )
15	10 12 13 14	syl3anc	|- ( ( ph /\ x e. A ) -> x <_ C )
16	15	ralrimiva	|- ( ph -> A. x e. A x <_ C )
17		brralrspcev	|- ( ( C e. RR /\ A. x e. A x <_ C ) -> E. y e. RR A. x e. A x <_ y )
18	2 16 17	syl2anc	|- ( ph -> E. y e. RR A. x e. A x <_ y )
19	8 4 18 5	suprubd	|- ( ph -> D <_ sup ( A , RR , < ) )