Theorem supsnti 6982

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 3612	. . 3 |- ( B e. A -> B e. { B } )
4	2, 3	syl 14	. 2 |- ( ph -> B e. { B } )
5		eqid 2170	. . . . . 6 |- B = B
6	1	ralrimivva 2552	. . . . . . 7 |- ( ph -> A. u e. A A. v e. A ( u = v <-> ( -. u R v /\ -. v R u ) ) )
7		eqeq1 2177	. . . . . . . . . 10 |- ( u = B -> ( u = v <-> B = v ) )
8		breq1 3992	. . . . . . . . . . . 12 |- ( u = B -> ( u R v <-> B R v ) )
9	8	notbid 662	. . . . . . . . . . 11 |- ( u = B -> ( -. u R v <-> -. B R v ) )
10		breq2 3993	. . . . . . . . . . . 12 |- ( u = B -> ( v R u <-> v R B ) )
11	10	notbid 662	. . . . . . . . . . 11 |- ( u = B -> ( -. v R u <-> -. v R B ) )
12	9, 11	anbi12d 470	. . . . . . . . . 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 2180	. . . . . . . . . 10 |- ( v = B -> ( B = v <-> B = B ) )
15		breq2 3993	. . . . . . . . . . . 12 |- ( v = B -> ( B R v <-> B R B ) )
16	15	notbid 662	. . . . . . . . . . 11 |- ( v = B -> ( -. B R v <-> -. B R B ) )
17		breq1 3992	. . . . . . . . . . . 12 |- ( v = B -> ( v R B <-> B R B ) )
18	17	notbid 662	. . . . . . . . . . 11 |- ( v = B -> ( -. v R B <-> -. B R B ) )
19	16, 18	anbi12d 470	. . . . . . . . . 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 2847	. . . . . . . 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 3601	. . . . . 6 |- ( x e. { B } -> x = B )
28	27	breq2d 4001	. . . . 5 |- ( x e. { B } -> ( B R x <-> B R B ) )
29	28	notbid 662	. . . 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 6981	1 |- ( ph -> sup ( { B } , A , R ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1348   e. wcel 2141   A.wral 2448   {csn 3583   class class class wbr 3989   supcsup 6959

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-riota 5809  df-sup 6961

This theorem is referenced by:  infsnti  7007