Theorem infminti 7004

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 527	. . 3 |- ( ( ph /\ ( y e. A /\ C R y ) ) -> C R y )
6		breq1 3992	. . . 4 |- ( z = C -> ( z R y <-> C R y ) )
7	6	rspcev 2834	. . 3 |- ( ( C e. B /\ C R y ) -> E. z e. B z R y )
8	4, 5, 7	syl2an2r 590	. 2 |- ( ( ph /\ ( y e. A /\ C R y ) ) -> E. z e. B z R y )
9	1, 2, 3, 8	eqinftid 6998	1 |- ( ph -> inf ( B , A , R ) = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   E.wrex 2449   class class class wbr 3989  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:  lbinf  8864  lcmgcdlem  12031  pilem3  13498  inffz  14101  taupi  14102