Theorem infminti 6992

Description: The smallest element of a set is its infimum. Note that the converse is not true; the infimum might not be an element of the set considered. (Contributed by Jim Kingdon, 18-Dec-2021.)

Hypotheses

Ref	Expression
infminti.ti	|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )
infminti.2	|- ( ph -> C e. A )
infminti.3	|- ( ph -> C e. B )
infminti.4	|- ( ( ph /\ y e. B ) -> -. y R C )

Assertion

Ref	Expression
infminti	|- ( ph -> inf ( B , A , R ) = C )

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

Proof of Theorem infminti

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		infminti.ti	. 2 |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )
2		infminti.2	. 2 |- ( ph -> C e. A )
3		infminti.4	. 2 |- ( ( ph /\ y e. B ) -> -. y R C )
4		infminti.3	. . 3 |- ( ph -> C e. B )
5		simprr 522	. . 3 |- ( ( ph /\ ( y e. A /\ C R y ) ) -> C R y )
6		breq1 3985	. . . 4 |- ( z = C -> ( z R y <-> C R y ) )
7	6	rspcev 2830	. . 3 |- ( ( C e. B /\ C R y ) -> E. z e. B z R y )
8	4, 5, 7	syl2an2r 585	. 2 |- ( ( ph /\ ( y e. A /\ C R y ) ) -> E. z e. B z R y )
9	1, 2, 3, 8	eqinftid 6986	1 |- ( ph -> inf ( B , A , R ) = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   E.wrex 2445   class class class wbr 3982  infcinf 6948

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-cnv 4612  df-iota 5153  df-riota 5798  df-sup 6949  df-inf 6950

This theorem is referenced by:  lbinf  8843  lcmgcdlem  12009  pilem3  13344  inffz  13948  taupi  13949