Theorem infminti 7028

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 531	. . 3 |- ( ( ph /\ ( y e. A /\ C R y ) ) -> C R y )
6		breq1 4008	. . . 4 |- ( z = C -> ( z R y <-> C R y ) )
7	6	rspcev 2843	. . 3 |- ( ( C e. B /\ C R y ) -> E. z e. B z R y )
8	4, 5, 7	syl2an2r 595	. 2 |- ( ( ph /\ ( y e. A /\ C R y ) ) -> E. z e. B z R y )
9	1, 2, 3, 8	eqinftid 7022	1 |- ( ph -> inf ( B , A , R ) = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   E.wrex 2456   class class class wbr 4005  infcinf 6984

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-cnv 4636  df-iota 5180  df-riota 5833  df-sup 6985  df-inf 6986

This theorem is referenced by:  lbinf  8907  lcmgcdlem  12079  pilem3  14243  inffz  14859  taupi  14860