Theorem sosng 4701

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 4315	. . 3 |- ( R Or { A } -> R Po { A } )
2		posng 4700	. . 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 615	. . . . . . . . 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 3612	. . . . . . . . . . 11 |- ( x e. { A } -> x = A )
8		elsni 3612	. . . . . . . . . . 11 |- ( y e. { A } -> y = A )
9	7, 8	breqan12d 4021	. . . . . . . . . 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 2551	. . . . . 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 2559	. . . . 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 4299	. . . 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 708   e. wcel 2148   A.wral 2455   _Vcvv 2739   {csn 3594   class class class wbr 4005   Po wpo 4296   Or wor 4297   Rel wrel 4633

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-sbc 2965  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-po 4298  df-iso 4299

This theorem is referenced by: (None)