Theorem infminti 7129

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 4047	. . . 4 |- ( z = C -> ( z R y <-> C R y ) )
7	6	rspcev 2877	. . 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 7123	1 |- ( ph -> inf ( B , A , R ) = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   E.wrex 2485   class class class wbr 4044  infcinf 7085

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-cnv 4683  df-iota 5232  df-riota 5899  df-sup 7086  df-inf 7087

This theorem is referenced by:  lbinf  9021  lcmgcdlem  12399  pilem3  15255  inffz  16011  taupi  16012