Theorem infsnti 7334

Description: The infimum of a singleton. (Contributed by Jim Kingdon, 19-Dec-2021.)

Hypotheses

Ref	Expression
infsnti.ti	|- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )
infsnti.b	|- ( ph -> B e. A )

Assertion

Ref	Expression
infsnti	|- ( ph -> inf ( { B } , A , R ) = B )

Distinct variable groups:   u, A, v   u, B, v   u, R, v   ph, u, v

Proof of Theorem infsnti

Step	Hyp	Ref	Expression
1		df-inf 7289	. 2 |- inf ( { B } , A , R ) = sup ( { B } , A , `' R )
2		infsnti.ti	. . . 4 |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )
3	2	cnvti 7323	. . 3 |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u `' R v /\ -. v `' R u ) ) )
4		infsnti.b	. . 3 |- ( ph -> B e. A )
5	3, 4	supsnti 7309	. 2 |- ( ph -> sup ( { B } , A , `' R ) = B )
6	1, 5	eqtrid 2279	1 |- ( ph -> inf ( { B } , A , R ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   {csn 3694   class class class wbr 4114   `'ccnv 4753   supcsup 7286  infcinf 7287

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-cnv 4762  df-iota 5317  df-riota 6011  df-sup 7288  df-inf 7289

This theorem is referenced by: (None)