Theorem inflbti 6826

Description: An infimum is a lower bound. See also infclti 6825 and infglbti 6827. (Contributed by Jim Kingdon, 18-Dec-2021.)

Hypotheses

Ref	Expression
infclti.ti	|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )
infclti.ex	|- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )

Assertion

Ref	Expression
inflbti	|- ( ph -> ( C e. B -> -. C Rinf ( B , A , R ) ) )

Distinct variable groups:   u, A, v, x, y, z   u, B, v, x, y, z   u, R, v, x, y, z   ph, u, v, x, y, z

Allowed substitution hints:   C( x, y, z, v, u)

Proof of Theorem inflbti

Step	Hyp	Ref	Expression
1		infclti.ti	. . . . . 6 |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )
2	1	cnvti 6821	. . . . 5 |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u `' R v /\ -. v `' R u ) ) )
3		infclti.ex	. . . . . 6 |- ( ph -> E. x e. A ( A. y e. B -. y R x /\ A. y e. A ( x R y -> E. z e. B z R y ) ) )
4	3	cnvinfex 6820	. . . . 5 |- ( 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 ) ) )
5	2, 4	supubti 6801	. . . 4 |- ( ph -> ( C e. B -> -. sup ( B , A , `' R ) `' R C ) )
6	5	imp 123	. . 3 |- ( ( ph /\ C e. B ) -> -. sup ( B , A , `' R ) `' R C )
7		df-inf 6787	. . . . . 6 |- inf ( B , A , R ) = sup ( B , A , `' R )
8	7	a1i 9	. . . . 5 |- ( ( ph /\ C e. B ) -> inf ( B , A , R ) = sup ( B , A , `' R ) )
9	8	breq2d 3887	. . . 4 |- ( ( ph /\ C e. B ) -> ( C Rinf ( B , A , R ) <-> C R sup ( B , A , `' R ) ) )
10	2, 4	supclti 6800	. . . . 5 |- ( ph -> sup ( B , A , `' R ) e. A )
11		brcnvg 4658	. . . . . 6 |- ( ( sup ( B , A , `' R ) e. A /\ C e. B ) -> ( sup ( B , A , `' R ) `' R C <-> C R sup ( B , A , `' R ) ) )
12	11	bicomd 140	. . . . 5 |- ( ( sup ( B , A , `' R ) e. A /\ C e. B ) -> ( C R sup ( B , A , `' R ) <-> sup ( B , A , `' R ) `' R C ) )
13	10, 12	sylan 279	. . . 4 |- ( ( ph /\ C e. B ) -> ( C R sup ( B , A , `' R ) <-> sup ( B , A , `' R ) `' R C ) )
14	9, 13	bitrd 187	. . 3 |- ( ( ph /\ C e. B ) -> ( C Rinf ( B , A , R ) <-> sup ( B , A , `' R ) `' R C ) )
15	6, 14	mtbird 639	. 2 |- ( ( ph /\ C e. B ) -> -. C Rinf ( B , A , R ) )
16	15	ex 114	1 |- ( ph -> ( C e. B -> -. C Rinf ( B , A , R ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1299   e. wcel 1448   A.wral 2375   E.wrex 2376   class class class wbr 3875   `'ccnv 4476   supcsup 6784  infcinf 6785

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-reu 2382  df-rmo 2383  df-rab 2384  df-v 2643  df-sbc 2863  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-cnv 4485  df-iota 5024  df-riota 5662  df-sup 6786  df-inf 6787

This theorem is referenced by:  zssinfcl  11436  infssuzledc  11438