Theorem supsnti 6418

Description: The supremum of a singleton. (Contributed by Jim Kingdon, 26-Nov-2021.)

Hypotheses

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

Assertion

Ref	Expression
supsnti	|- ( ph -> sup ( { B } , A , R ) = B )

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

Proof of Theorem supsnti

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		supsnti.ti	. 2 |- ( ( ph /\ ( u e. A /\ v e. A ) ) -> ( u = v <-> ( -. u R v /\ -. v R u ) ) )
2		supsnti.b	. 2 |- ( ph -> B e. A )
3		snidg 3423	. . 3 |- ( B e. A -> B e. { B } )
4	2, 3	syl 14	. 2 |- ( ph -> B e. { B } )
5		eqid 2081	. . . . . 6 |- B = B
6	1	ralrimivva 2443	. . . . . . 7 |- ( ph -> A. u e. A A. v e. A ( u = v <-> ( -. u R v /\ -. v R u ) ) )
7		eqeq1 2087	. . . . . . . . . 10 |- ( u = B -> ( u = v <-> B = v ) )
8		breq1 3788	. . . . . . . . . . . 12 |- ( u = B -> ( u R v <-> B R v ) )
9	8	notbid 624	. . . . . . . . . . 11 |- ( u = B -> ( -. u R v <-> -. B R v ) )
10		breq2 3789	. . . . . . . . . . . 12 |- ( u = B -> ( v R u <-> v R B ) )
11	10	notbid 624	. . . . . . . . . . 11 |- ( u = B -> ( -. v R u <-> -. v R B ) )
12	9, 11	anbi12d 456	. . . . . . . . . 10 |- ( u = B -> ( ( -. u R v /\ -. v R u ) <-> ( -. B R v /\ -. v R B ) ) )
13	7, 12	bibi12d 233	. . . . . . . . 9 |- ( u = B -> ( ( u = v <-> ( -. u R v /\ -. v R u ) ) <-> ( B = v <-> ( -. B R v /\ -. v R B ) ) ) )
14		eqeq2 2090	. . . . . . . . . 10 |- ( v = B -> ( B = v <-> B = B ) )
15		breq2 3789	. . . . . . . . . . . 12 |- ( v = B -> ( B R v <-> B R B ) )
16	15	notbid 624	. . . . . . . . . . 11 |- ( v = B -> ( -. B R v <-> -. B R B ) )
17		breq1 3788	. . . . . . . . . . . 12 |- ( v = B -> ( v R B <-> B R B ) )
18	17	notbid 624	. . . . . . . . . . 11 |- ( v = B -> ( -. v R B <-> -. B R B ) )
19	16, 18	anbi12d 456	. . . . . . . . . 10 |- ( v = B -> ( ( -. B R v /\ -. v R B ) <-> ( -. B R B /\ -. B R B ) ) )
20	14, 19	bibi12d 233	. . . . . . . . 9 |- ( v = B -> ( ( B = v <-> ( -. B R v /\ -. v R B ) ) <-> ( B = B <-> ( -. B R B /\ -. B R B ) ) ) )
21	13, 20	rspc2v 2713	. . . . . . . 8 |- ( ( B e. A /\ B e. A ) -> ( A. u e. A A. v e. A ( u = v <-> ( -. u R v /\ -. v R u ) ) -> ( B = B <-> ( -. B R B /\ -. B R B ) ) ) )
22	2, 2, 21	syl2anc 403	. . . . . . 7 |- ( ph -> ( A. u e. A A. v e. A ( u = v <-> ( -. u R v /\ -. v R u ) ) -> ( B = B <-> ( -. B R B /\ -. B R B ) ) ) )
23	6, 22	mpd 13	. . . . . 6 |- ( ph -> ( B = B <-> ( -. B R B /\ -. B R B ) ) )
24	5, 23	mpbii 146	. . . . 5 |- ( ph -> ( -. B R B /\ -. B R B ) )
25	24	simpld 110	. . . 4 |- ( ph -> -. B R B )
26	25	adantr 270	. . 3 |- ( ( ph /\ x e. { B } ) -> -. B R B )
27		elsni 3416	. . . . . 6 |- ( x e. { B } -> x = B )
28	27	breq2d 3797	. . . . 5 |- ( x e. { B } -> ( B R x <-> B R B ) )
29	28	notbid 624	. . . 4 |- ( x e. { B } -> ( -. B R x <-> -. B R B ) )
30	29	adantl 271	. . 3 |- ( ( ph /\ x e. { B } ) -> ( -. B R x <-> -. B R B ) )
31	26, 30	mpbird 165	. 2 |- ( ( ph /\ x e. { B } ) -> -. B R x )
32	1, 2, 4, 31	supmaxti 6417	1 |- ( ph -> sup ( { B } , A , R ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348   {csn 3398   class class class wbr 3785   supcsup 6395

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-iota 4887  df-riota 5488  df-sup 6397

This theorem is referenced by:  infsnti  6443