Theorem supsnti 6970

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 3605	. . 3 |- ( B e. A -> B e. { B } )
4	2, 3	syl 14	. 2 |- ( ph -> B e. { B } )
5		eqid 2165	. . . . . 6 |- B = B
6	1	ralrimivva 2548	. . . . . . 7 |- ( ph -> A. u e. A A. v e. A ( u = v <-> ( -. u R v /\ -. v R u ) ) )
7		eqeq1 2172	. . . . . . . . . 10 |- ( u = B -> ( u = v <-> B = v ) )
8		breq1 3985	. . . . . . . . . . . 12 |- ( u = B -> ( u R v <-> B R v ) )
9	8	notbid 657	. . . . . . . . . . 11 |- ( u = B -> ( -. u R v <-> -. B R v ) )
10		breq2 3986	. . . . . . . . . . . 12 |- ( u = B -> ( v R u <-> v R B ) )
11	10	notbid 657	. . . . . . . . . . 11 |- ( u = B -> ( -. v R u <-> -. v R B ) )
12	9, 11	anbi12d 465	. . . . . . . . . 10 |- ( u = B -> ( ( -. u R v /\ -. v R u ) <-> ( -. B R v /\ -. v R B ) ) )
13	7, 12	bibi12d 234	. . . . . . . . 9 |- ( u = B -> ( ( u = v <-> ( -. u R v /\ -. v R u ) ) <-> ( B = v <-> ( -. B R v /\ -. v R B ) ) ) )
14		eqeq2 2175	. . . . . . . . . 10 |- ( v = B -> ( B = v <-> B = B ) )
15		breq2 3986	. . . . . . . . . . . 12 |- ( v = B -> ( B R v <-> B R B ) )
16	15	notbid 657	. . . . . . . . . . 11 |- ( v = B -> ( -. B R v <-> -. B R B ) )
17		breq1 3985	. . . . . . . . . . . 12 |- ( v = B -> ( v R B <-> B R B ) )
18	17	notbid 657	. . . . . . . . . . 11 |- ( v = B -> ( -. v R B <-> -. B R B ) )
19	16, 18	anbi12d 465	. . . . . . . . . 10 |- ( v = B -> ( ( -. B R v /\ -. v R B ) <-> ( -. B R B /\ -. B R B ) ) )
20	14, 19	bibi12d 234	. . . . . . . . 9 |- ( v = B -> ( ( B = v <-> ( -. B R v /\ -. v R B ) ) <-> ( B = B <-> ( -. B R B /\ -. B R B ) ) ) )
21	13, 20	rspc2v 2843	. . . . . . . 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 409	. . . . . . 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 147	. . . . 5 |- ( ph -> ( -. B R B /\ -. B R B ) )
25	24	simpld 111	. . . 4 |- ( ph -> -. B R B )
26	25	adantr 274	. . 3 |- ( ( ph /\ x e. { B } ) -> -. B R B )
27		elsni 3594	. . . . . 6 |- ( x e. { B } -> x = B )
28	27	breq2d 3994	. . . . 5 |- ( x e. { B } -> ( B R x <-> B R B ) )
29	28	notbid 657	. . . 4 |- ( x e. { B } -> ( -. B R x <-> -. B R B ) )
30	29	adantl 275	. . 3 |- ( ( ph /\ x e. { B } ) -> ( -. B R x <-> -. B R B ) )
31	26, 30	mpbird 166	. 2 |- ( ( ph /\ x e. { B } ) -> -. B R x )
32	1, 2, 4, 31	supmaxti 6969	1 |- ( ph -> sup ( { B } , A , R ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   A.wral 2444   {csn 3576   class class class wbr 3982   supcsup 6947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-un 3120  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-riota 5798  df-sup 6949

This theorem is referenced by:  infsnti  6995