Theorem supgtoreq 8376

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.)

Hypotheses

Ref	Expression
supgtoreq.1	|- ( ph -> R Or A )
supgtoreq.2	|- ( ph -> B C_ A )
supgtoreq.3	|- ( ph -> B e. Fin )
supgtoreq.4	|- ( ph -> C e. B )
supgtoreq.5	|- ( ph -> S = sup ( B , A , R ) )

Assertion

Ref	Expression
supgtoreq	|- ( ph -> ( C R S \/ C = S ) )

Proof of Theorem supgtoreq

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		supgtoreq.4	. . . . 5 |- ( ph -> C e. B )
2		supgtoreq.1	. . . . . 6 |- ( ph -> R Or A )
3		supgtoreq.2	. . . . . . 7 |- ( ph -> B C_ A )
4		supgtoreq.3	. . . . . . . 8 |- ( ph -> B e. Fin )
5		ne0i 3921	. . . . . . . . 9 |- ( C e. B -> B =/= (/) )
6	1, 5	syl 17	. . . . . . . 8 |- ( ph -> B =/= (/) )
7		fisup2g 8374	. . . . . . . 8 |- ( ( R Or A /\ ( B e. Fin /\ B =/= (/) /\ B C_ A ) ) -> E. x e. B ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )
8	2, 4, 6, 3, 7	syl13anc 1328	. . . . . . 7 |- ( ph -> E. x e. B ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )
9		ssrexv 3667	. . . . . . 7 |- ( B C_ A -> ( E. x e. B ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) -> E. x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) ) )
10	3, 8, 9	sylc 65	. . . . . 6 |- ( ph -> E. x e. A ( A. y e. B -. x R y /\ A. y e. A ( y R x -> E. z e. B y R z ) ) )
11	2, 10	supub 8365	. . . . 5 |- ( ph -> ( C e. B -> -. sup ( B , A , R ) R C ) )
12	1, 11	mpd 15	. . . 4 |- ( ph -> -. sup ( B , A , R ) R C )
13		supgtoreq.5	. . . . 5 |- ( ph -> S = sup ( B , A , R ) )
14	13	breq1d 4663	. . . 4 |- ( ph -> ( S R C <-> sup ( B , A , R ) R C ) )
15	12, 14	mtbird 315	. . 3 |- ( ph -> -. S R C )
16		fisupcl 8375	. . . . . . . 8 |- ( ( R Or A /\ ( B e. Fin /\ B =/= (/) /\ B C_ A ) ) -> sup ( B , A , R ) e. B )
17	2, 4, 6, 3, 16	syl13anc 1328	. . . . . . 7 |- ( ph -> sup ( B , A , R ) e. B )
18	3, 17	sseldd 3604	. . . . . 6 |- ( ph -> sup ( B , A , R ) e. A )
19	13, 18	eqeltrd 2701	. . . . 5 |- ( ph -> S e. A )
20	3, 1	sseldd 3604	. . . . 5 |- ( ph -> C e. A )
21		sotric 5061	. . . . 5 |- ( ( R Or A /\ ( S e. A /\ C e. A ) ) -> ( S R C <-> -. ( S = C \/ C R S ) ) )
22	2, 19, 20, 21	syl12anc 1324	. . . 4 |- ( ph -> ( S R C <-> -. ( S = C \/ C R S ) ) )
23		orcom 402	. . . . . 6 |- ( ( S = C \/ C R S ) <-> ( C R S \/ S = C ) )
24		eqcom 2629	. . . . . . 7 |- ( S = C <-> C = S )
25	24	orbi2i 541	. . . . . 6 |- ( ( C R S \/ S = C ) <-> ( C R S \/ C = S ) )
26	23, 25	bitri 264	. . . . 5 |- ( ( S = C \/ C R S ) <-> ( C R S \/ C = S ) )
27	26	notbii 310	. . . 4 |- ( -. ( S = C \/ C R S ) <-> -. ( C R S \/ C = S ) )
28	22, 27	syl6rbb 277	. . 3 |- ( ph -> ( -. ( C R S \/ C = S ) <-> S R C ) )
29	15, 28	mtbird 315	. 2 |- ( ph -> -. -. ( C R S \/ C = S ) )
30	29	notnotrd 128	1 |- ( ph -> ( C R S \/ C = S ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   C_ wss 3574   (/)c0 3915   class class class wbr 4653   Or wor 5034   Fincfn 7955   supcsup 8346

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

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-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  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-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  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-om 7066  df-1o 7560  df-er 7742  df-en 7956  df-fin 7959  df-sup 8348

This theorem is referenced by:  infltoreq  8408  supfirege  11009