Theorem sosng 4431

Description: Strict linear ordering on a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)

Assertion

Ref	Expression
sosng	|- ( ( Rel R /\ A e. _V ) -> ( R Or { A } <-> -. A R A ) )

Proof of Theorem sosng

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sopo 4068	. . 3 |- ( R Or { A } -> R Po { A } )
2		posng 4430	. . 3 |- ( ( Rel R /\ A e. _V ) -> ( R Po { A } <-> -. A R A ) )
3	1, 2	syl5ib 152	. 2 |- ( ( Rel R /\ A e. _V ) -> ( R Or { A } -> -. A R A ) )
4	2	biimpar 291	. . . 4 |- ( ( ( Rel R /\ A e. _V ) /\ -. A R A ) -> R Po { A } )
5		ax-in2 577	. . . . . . . . 9 |- ( -. A R A -> ( A R A -> ( x R z \/ z R y ) ) )
6	5	adantr 270	. . . . . . . 8 |- ( ( -. A R A /\ ( x e. { A } /\ y e. { A } ) ) -> ( A R A -> ( x R z \/ z R y ) ) )
7		elsni 3416	. . . . . . . . . . 11 |- ( x e. { A } -> x = A )
8		elsni 3416	. . . . . . . . . . 11 |- ( y e. { A } -> y = A )
9	7, 8	breqan12d 3800	. . . . . . . . . 10 |- ( ( x e. { A } /\ y e. { A } ) -> ( x R y <-> A R A ) )
10	9	imbi1d 229	. . . . . . . . 9 |- ( ( x e. { A } /\ y e. { A } ) -> ( ( x R y -> ( x R z \/ z R y ) ) <-> ( A R A -> ( x R z \/ z R y ) ) ) )
11	10	adantl 271	. . . . . . . 8 |- ( ( -. A R A /\ ( x e. { A } /\ y e. { A } ) ) -> ( ( x R y -> ( x R z \/ z R y ) ) <-> ( A R A -> ( x R z \/ z R y ) ) ) )
12	6, 11	mpbird 165	. . . . . . 7 |- ( ( -. A R A /\ ( x e. { A } /\ y e. { A } ) ) -> ( x R y -> ( x R z \/ z R y ) ) )
13	12	ralrimivw 2435	. . . . . 6 |- ( ( -. A R A /\ ( x e. { A } /\ y e. { A } ) ) -> A. z e. { A } ( x R y -> ( x R z \/ z R y ) ) )
14	13	ralrimivva 2443	. . . . 5 |- ( -. A R A -> A. x e. { A } A. y e. { A } A. z e. { A } ( x R y -> ( x R z \/ z R y ) ) )
15	14	adantl 271	. . . 4 |- ( ( ( Rel R /\ A e. _V ) /\ -. A R A ) -> A. x e. { A } A. y e. { A } A. z e. { A } ( x R y -> ( x R z \/ z R y ) ) )
16		df-iso 4052	. . . 4 |- ( R Or { A } <-> ( R Po { A } /\ A. x e. { A } A. y e. { A } A. z e. { A } ( x R y -> ( x R z \/ z R y ) ) ) )
17	4, 15, 16	sylanbrc 408	. . 3 |- ( ( ( Rel R /\ A e. _V ) /\ -. A R A ) -> R Or { A } )
18	17	ex 113	. 2 |- ( ( Rel R /\ A e. _V ) -> ( -. A R A -> R Or { A } ) )
19	3, 18	impbid 127	1 |- ( ( Rel R /\ A e. _V ) -> ( R Or { A } <-> -. A R A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661   e. wcel 1433   A.wral 2348   _Vcvv 2601   {csn 3398   class class class wbr 3785   Po wpo 4049   Or wor 4050   Rel wrel 4368

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-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603  df-sbc 2816  df-un 2977  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-po 4051  df-iso 4052

This theorem is referenced by: (None)