Theorem sosng 4711

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 4325	. . 3 |- ( R Or { A } -> R Po { A } )
2		posng 4710	. . 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 3622	. . . . . . . . . . 11 |- ( x e. { A } -> x = A )
8		elsni 3622	. . . . . . . . . . 11 |- ( y e. { A } -> y = A )
9	7, 8	breqan12d 4031	. . . . . . . . . 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 2561	. . . . . 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 2569	. . . . 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 4309	. . . 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 2158   A.wral 2465   _Vcvv 2749   {csn 3604   class class class wbr 4015   Po wpo 4306   Or wor 4307   Rel wrel 4643

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2169

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-v 2751  df-sbc 2975  df-un 3145  df-sn 3610  df-pr 3611  df-op 3613  df-br 4016  df-po 4308  df-iso 4309

This theorem is referenced by: (None)