Theorem sosng 4748

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 4360	. . 3 |- ( R Or { A } -> R Po { A } )
2		posng 4747	. . 3 |- ( ( Rel R /\ A e. _V ) -> ( R Po { A } <-> -. A R A ) )
3	1, 2	imbitrid 154	. 2 |- ( ( Rel R /\ A e. _V ) -> ( R Or { A } -> -. A R A ) )
4	2	biimpar 297	. . . 4 |- ( ( ( Rel R /\ A e. _V ) /\ -. A R A ) -> R Po { A } )
5		ax-in2 616	. . . . . . . . 9 |- ( -. A R A -> ( A R A -> ( x R z \/ z R y ) ) )
6	5	adantr 276	. . . . . . . 8 |- ( ( -. A R A /\ ( x e. { A } /\ y e. { A } ) ) -> ( A R A -> ( x R z \/ z R y ) ) )
7		elsni 3651	. . . . . . . . . . 11 |- ( x e. { A } -> x = A )
8		elsni 3651	. . . . . . . . . . 11 |- ( y e. { A } -> y = A )
9	7, 8	breqan12d 4060	. . . . . . . . . 10 |- ( ( x e. { A } /\ y e. { A } ) -> ( x R y <-> A R A ) )
10	9	imbi1d 231	. . . . . . . . 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 277	. . . . . . . 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 167	. . . . . . 7 |- ( ( -. A R A /\ ( x e. { A } /\ y e. { A } ) ) -> ( x R y -> ( x R z \/ z R y ) ) )
13	12	ralrimivw 2580	. . . . . 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 2588	. . . . 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 277	. . . 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 4344	. . . 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 417	. . 3 |- ( ( ( Rel R /\ A e. _V ) /\ -. A R A ) -> R Or { A } )
18	17	ex 115	. 2 |- ( ( Rel R /\ A e. _V ) -> ( -. A R A -> R Or { A } ) )
19	3, 18	impbid 129	1 |- ( ( Rel R /\ A e. _V ) -> ( R Or { A } <-> -. A R A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   e. wcel 2176   A.wral 2484   _Vcvv 2772   {csn 3633   class class class wbr 4044   Po wpo 4341   Or wor 4342   Rel wrel 4680

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-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-v 2774  df-sbc 2999  df-un 3170  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-po 4343  df-iso 4344

This theorem is referenced by: (None)