Theorem inflbti 7001

Description: An infimum is a lower bound. See also infclti 7000 and infglbti 7002. (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 6996	. . . . 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 6995	. . . . 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 6976	. . . 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 6962	. . . . . 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 4001	. . . 4 |- ( ( ph /\ C e. B ) -> ( C Rinf ( B , A , R ) <-> C R sup ( B , A , `' R ) ) )
10	2, 4	supclti 6975	. . . . 5 |- ( ph -> sup ( B , A , `' R ) e. A )
11		brcnvg 4792	. . . . . 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 281	. . . 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 668	. 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 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   class class class wbr 3989   `'ccnv 4610   supcsup 6959  infcinf 6960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-cnv 4619  df-iota 5160  df-riota 5809  df-sup 6961  df-inf 6962

This theorem is referenced by:  infregelbex  9557  zssinfcl  11903  infssuzledc  11905