Theorem supsnti 7107

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 3662	. . 3 |- ( B e. A -> B e. { B } )
4	2, 3	syl 14	. 2 |- ( ph -> B e. { B } )
5		eqid 2205	. . . . . 6 |- B = B
6	1	ralrimivva 2588	. . . . . . 7 |- ( ph -> A. u e. A A. v e. A ( u = v <-> ( -. u R v /\ -. v R u ) ) )
7		eqeq1 2212	. . . . . . . . . 10 |- ( u = B -> ( u = v <-> B = v ) )
8		breq1 4047	. . . . . . . . . . . 12 |- ( u = B -> ( u R v <-> B R v ) )
9	8	notbid 669	. . . . . . . . . . 11 |- ( u = B -> ( -. u R v <-> -. B R v ) )
10		breq2 4048	. . . . . . . . . . . 12 |- ( u = B -> ( v R u <-> v R B ) )
11	10	notbid 669	. . . . . . . . . . 11 |- ( u = B -> ( -. v R u <-> -. v R B ) )
12	9, 11	anbi12d 473	. . . . . . . . . 10 |- ( u = B -> ( ( -. u R v /\ -. v R u ) <-> ( -. B R v /\ -. v R B ) ) )
13	7, 12	bibi12d 235	. . . . . . . . 9 |- ( u = B -> ( ( u = v <-> ( -. u R v /\ -. v R u ) ) <-> ( B = v <-> ( -. B R v /\ -. v R B ) ) ) )
14		eqeq2 2215	. . . . . . . . . 10 |- ( v = B -> ( B = v <-> B = B ) )
15		breq2 4048	. . . . . . . . . . . 12 |- ( v = B -> ( B R v <-> B R B ) )
16	15	notbid 669	. . . . . . . . . . 11 |- ( v = B -> ( -. B R v <-> -. B R B ) )
17		breq1 4047	. . . . . . . . . . . 12 |- ( v = B -> ( v R B <-> B R B ) )
18	17	notbid 669	. . . . . . . . . . 11 |- ( v = B -> ( -. v R B <-> -. B R B ) )
19	16, 18	anbi12d 473	. . . . . . . . . 10 |- ( v = B -> ( ( -. B R v /\ -. v R B ) <-> ( -. B R B /\ -. B R B ) ) )
20	14, 19	bibi12d 235	. . . . . . . . 9 |- ( v = B -> ( ( B = v <-> ( -. B R v /\ -. v R B ) ) <-> ( B = B <-> ( -. B R B /\ -. B R B ) ) ) )
21	13, 20	rspc2v 2890	. . . . . . . 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 411	. . . . . . 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 148	. . . . 5 |- ( ph -> ( -. B R B /\ -. B R B ) )
25	24	simpld 112	. . . 4 |- ( ph -> -. B R B )
26	25	adantr 276	. . 3 |- ( ( ph /\ x e. { B } ) -> -. B R B )
27		elsni 3651	. . . . . 6 |- ( x e. { B } -> x = B )
28	27	breq2d 4056	. . . . 5 |- ( x e. { B } -> ( B R x <-> B R B ) )
29	28	notbid 669	. . . 4 |- ( x e. { B } -> ( -. B R x <-> -. B R B ) )
30	29	adantl 277	. . 3 |- ( ( ph /\ x e. { B } ) -> ( -. B R x <-> -. B R B ) )
31	26, 30	mpbird 167	. 2 |- ( ( ph /\ x e. { B } ) -> -. B R x )
32	1, 2, 4, 31	supmaxti 7106	1 |- ( ph -> sup ( { B } , A , R ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1373   e. wcel 2176   A.wral 2484   {csn 3633   class class class wbr 4044   supcsup 7084

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-riota 5899  df-sup 7086

This theorem is referenced by:  infsnti  7132