Metamath Proof Explorer


Theorem supgtoreq

Description: The supremum of a finite set is greater than or equal to all the elements of the set. (Contributed by AV, 1-Oct-2019)

Ref Expression
Hypotheses supgtoreq.1 φ R Or A
supgtoreq.2 φ B A
supgtoreq.3 φ B Fin
supgtoreq.4 φ C B
supgtoreq.5 φ S = sup B A R
Assertion supgtoreq φ C R S C = S

Proof

Step Hyp Ref Expression
1 supgtoreq.1 φ R Or A
2 supgtoreq.2 φ B A
3 supgtoreq.3 φ B Fin
4 supgtoreq.4 φ C B
5 supgtoreq.5 φ S = sup B A R
6 4 ne0d φ B
7 fisup2g R Or A B Fin B B A x B y B ¬ x R y y A y R x z B y R z
8 1 3 6 2 7 syl13anc φ x B y B ¬ x R y y A y R x z B y R z
9 ssrexv B A x B y B ¬ x R y y A y R x z B y R z x A y B ¬ x R y y A y R x z B y R z
10 2 8 9 sylc φ x A y B ¬ x R y y A y R x z B y R z
11 1 10 supub φ C B ¬ sup B A R R C
12 4 11 mpd φ ¬ sup B A R R C
13 5 12 eqnbrtrd φ ¬ S R C
14 fisupcl R Or A B Fin B B A sup B A R B
15 1 3 6 2 14 syl13anc φ sup B A R B
16 2 15 sseldd φ sup B A R A
17 5 16 eqeltrd φ S A
18 2 4 sseldd φ C A
19 sotric R Or A S A C A S R C ¬ S = C C R S
20 1 17 18 19 syl12anc φ S R C ¬ S = C C R S
21 orcom S = C C R S C R S S = C
22 eqcom S = C C = S
23 22 orbi2i C R S S = C C R S C = S
24 21 23 bitri S = C C R S C R S C = S
25 24 notbii ¬ S = C C R S ¬ C R S C = S
26 20 25 bitr2di φ ¬ C R S C = S S R C
27 13 26 mtbird φ ¬ ¬ C R S C = S
28 27 notnotrd φ C R S C = S