Theorem inflbti 7317

Description: An infimum is a lower bound. See also infclti 7316 and infglbti 7318. (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 7312	. . . . 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 7311	. . . . 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 7292	. . . 4 |- ( ph -> ( C e. B -> -. sup ( B , A , `' R ) `' R C ) )
6	5	imp 124	. . 3 |- ( ( ph /\ C e. B ) -> -. sup ( B , A , `' R ) `' R C )
7		df-inf 7278	. . . . . 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 4123	. . . 4 |- ( ( ph /\ C e. B ) -> ( C Rinf ( B , A , R ) <-> C R sup ( B , A , `' R ) ) )
10	2, 4	supclti 7291	. . . . 5 |- ( ph -> sup ( B , A , `' R ) e. A )
11		brcnvg 4938	. . . . . 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 141	. . . . 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 283	. . . 4 |- ( ( ph /\ C e. B ) -> ( C R sup ( B , A , `' R ) <-> sup ( B , A , `' R ) `' R C ) )
14	9, 13	bitrd 188	. . 3 |- ( ( ph /\ C e. B ) -> ( C Rinf ( B , A , R ) <-> sup ( B , A , `' R ) `' R C ) )
15	6, 14	mtbird 680	. 2 |- ( ( ph /\ C e. B ) -> -. C Rinf ( B , A , R ) )
16	15	ex 115	1 |- ( ph -> ( C e. B -> -. C Rinf ( B , A , R ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522   E.wrex 2523   class class class wbr 4111   `'ccnv 4750   supcsup 7275  infcinf 7276

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-cnv 4759  df-iota 5314  df-riota 6005  df-sup 7277  df-inf 7278

This theorem is referenced by:  infregelbex  9933  zssinfcl  10596  infssuzledc  10598