Theorem infmin 8400

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 AV, 3-Sep-2020.)

Hypotheses

Ref	Expression
infmin.1	|- ( ph -> R Or A )
infmin.2	|- ( ph -> C e. A )
infmin.3	|- ( ph -> C e. B )
infmin.4	|- ( ( ph /\ y e. B ) -> -. y R C )

Assertion

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

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

Proof of Theorem infmin

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		infmin.1	. 2 |- ( ph -> R Or A )
2		infmin.2	. 2 |- ( ph -> C e. A )
3		infmin.4	. 2 |- ( ( ph /\ y e. B ) -> -. y R C )
4		infmin.3	. . . 4 |- ( ph -> C e. B )
5	4	adantr 481	. . 3 |- ( ( ph /\ ( y e. A /\ C R y ) ) -> C e. B )
6		simprr 796	. . 3 |- ( ( ph /\ ( y e. A /\ C R y ) ) -> C R y )
7		breq1 4656	. . . 4 |- ( z = C -> ( z R y <-> C R y ) )
8	7	rspcev 3309	. . 3 |- ( ( C e. B /\ C R y ) -> E. z e. B z R y )
9	5, 6, 8	syl2anc 693	. 2 |- ( ( ph /\ ( y e. A /\ C R y ) ) -> E. z e. B z R y )
10	1, 2, 3, 9	eqinfd 8391	1 |- ( ph -> inf ( B , A , R ) = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   class class class wbr 4653   Or wor 5034  infcinf 8347

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-cnv 5122  df-iota 5851  df-riota 6611  df-sup 8348  df-inf 8349

This theorem is referenced by:  infpr  8409  lbinf  10976  uzinfi  11768  lcmgcdlem  15319  ramcl2lem  15713  oms0  30359  ballotlemirc  30593  inffz  31614