Theorem supsnti 7335

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 3734	. . 3 |- ( B e. A -> B e. { B } )
4	2, 3	syl 14	. 2 |- ( ph -> B e. { B } )
5		eqid 2238	. . . . . 6 |- B = B
6	1	ralrimivva 2632	. . . . . . 7 |- ( ph -> A. u e. A A. v e. A ( u = v <-> ( -. u R v /\ -. v R u ) ) )
7		eqeq1 2245	. . . . . . . . . 10 |- ( u = B -> ( u = v <-> B = v ) )
8		breq1 4128	. . . . . . . . . . . 12 |- ( u = B -> ( u R v <-> B R v ) )
9	8	notbid 677	. . . . . . . . . . 11 |- ( u = B -> ( -. u R v <-> -. B R v ) )
10		breq2 4129	. . . . . . . . . . . 12 |- ( u = B -> ( v R u <-> v R B ) )
11	10	notbid 677	. . . . . . . . . . 11 |- ( u = B -> ( -. v R u <-> -. v R B ) )
12	9, 11	anbi12d 477	. . . . . . . . . 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 2248	. . . . . . . . . 10 |- ( v = B -> ( B = v <-> B = B ) )
15		breq2 4129	. . . . . . . . . . . 12 |- ( v = B -> ( B R v <-> B R B ) )
16	15	notbid 677	. . . . . . . . . . 11 |- ( v = B -> ( -. B R v <-> -. B R B ) )
17		breq1 4128	. . . . . . . . . . . 12 |- ( v = B -> ( v R B <-> B R B ) )
18	17	notbid 677	. . . . . . . . . . 11 |- ( v = B -> ( -. v R B <-> -. B R B ) )
19	16, 18	anbi12d 477	. . . . . . . . . 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 2943	. . . . . . . 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 415	. . . . . . 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 3723	. . . . . 6 |- ( x e. { B } -> x = B )
28	27	breq2d 4137	. . . . 5 |- ( x e. { B } -> ( B R x <-> B R B ) )
29	28	notbid 677	. . . 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 7334	1 |- ( ph -> sup ( { B } , A , R ) = B )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   A.wral 2528   {csn 3705   class class class wbr 4125   supcsup 7312

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-un 3224  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-riota 6028  df-sup 7314

This theorem is referenced by:  infsnti  7360