Theorem infltoreq 8408

Description: The infimum of a finite set is less than or equal to all the elements of the set. (Contributed by AV, 4-Sep-2020.)

Hypotheses

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

Assertion

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

Proof of Theorem infltoreq

Step	Hyp	Ref	Expression
1		infltoreq.1	. . . 4 |- ( ph -> R Or A )
2		cnvso 5674	. . . 4 |- ( R Or A <-> `' R Or A )
3	1, 2	sylib 208	. . 3 |- ( ph -> `' R Or A )
4		infltoreq.2	. . 3 |- ( ph -> B C_ A )
5		infltoreq.3	. . 3 |- ( ph -> B e. Fin )
6		infltoreq.4	. . 3 |- ( ph -> C e. B )
7		infltoreq.5	. . . 4 |- ( ph -> S = inf ( B , A , R ) )
8		df-inf 8349	. . . 4 |- inf ( B , A , R ) = sup ( B , A , `' R )
9	7, 8	syl6eq 2672	. . 3 |- ( ph -> S = sup ( B , A , `' R ) )
10	3, 4, 5, 6, 9	supgtoreq 8376	. 2 |- ( ph -> ( C `' R S \/ C = S ) )
11		ne0i 3921	. . . . . . 7 |- ( C e. B -> B =/= (/) )
12	6, 11	syl 17	. . . . . 6 |- ( ph -> B =/= (/) )
13		fiinfcl 8407	. . . . . 6 |- ( ( R Or A /\ ( B e. Fin /\ B =/= (/) /\ B C_ A ) ) -> inf ( B , A , R ) e. B )
14	1, 5, 12, 4, 13	syl13anc 1328	. . . . 5 |- ( ph -> inf ( B , A , R ) e. B )
15	7, 14	eqeltrd 2701	. . . 4 |- ( ph -> S e. B )
16		brcnvg 5303	. . . . 5 |- ( ( C e. B /\ S e. B ) -> ( C `' R S <-> S R C ) )
17	16	bicomd 213	. . . 4 |- ( ( C e. B /\ S e. B ) -> ( S R C <-> C `' R S ) )
18	6, 15, 17	syl2anc 693	. . 3 |- ( ph -> ( S R C <-> C `' R S ) )
19	18	orbi1d 739	. 2 |- ( ph -> ( ( S R C \/ C = S ) <-> ( C `' R S \/ C = S ) ) )
20	10, 19	mpbird 247	1 |- ( ph -> ( S R C \/ C = S ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   C_ wss 3574   (/)c0 3915   class class class wbr 4653   Or wor 5034   `'ccnv 5113   Fincfn 7955   supcsup 8346  infcinf 8347

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  df-inf 8349

This theorem is referenced by: (None)