Theorem infminti 7357

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 537	. . 3 |- ( ( ph /\ ( y e. A /\ C R y ) ) -> C R y )
6		breq1 4128	. . . 4 |- ( z = C -> ( z R y <-> C R y ) )
7	6	rspcev 2929	. . 3 |- ( ( C e. B /\ C R y ) -> E. z e. B z R y )
8	4, 5, 7	syl2an2r 603	. 2 |- ( ( ph /\ ( y e. A /\ C R y ) ) -> E. z e. B z R y )
9	1, 2, 3, 8	eqinftid 7351	1 |- ( ph -> inf ( B , A , R ) = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   E.wrex 2529   class class class wbr 4125  infcinf 7313

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-cnv 4777  df-iota 5332  df-riota 6028  df-sup 7314  df-inf 7315

This theorem is referenced by:  lbinf  9268  lcmgcdlem  12833  ballotfilemirc  13253  pilem3  15807  inffz  17027  taupi  17028