Theorem sosng 4737

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 4349	. . 3 |- ( R Or { A } -> R Po { A } )
2		posng 4736	. . 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 3641	. . . . . . . . . . 11 |- ( x e. { A } -> x = A )
8		elsni 3641	. . . . . . . . . . 11 |- ( y e. { A } -> y = A )
9	7, 8	breqan12d 4050	. . . . . . . . . 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 2571	. . . . . 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 2579	. . . . 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 4333	. . . 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 709   e. wcel 2167   A.wral 2475   _Vcvv 2763   {csn 3623   class class class wbr 4034   Po wpo 4330   Or wor 4331   Rel wrel 4669

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-v 2765  df-sbc 2990  df-un 3161  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-po 4332  df-iso 4333

This theorem is referenced by: (None)