Theorem sosn 5188

Description: Strict ordering on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)

Assertion

Ref	Expression
sosn	|- ( Rel R -> ( R Or { A } <-> -. A R A ) )

Proof of Theorem sosn

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

Step	Hyp	Ref	Expression
1		elsni 4194	. . . . . 6 |- ( x e. { A } -> x = A )
2		elsni 4194	. . . . . . 7 |- ( y e. { A } -> y = A )
3	2	eqcomd 2628	. . . . . 6 |- ( y e. { A } -> A = y )
4	1, 3	sylan9eq 2676	. . . . 5 |- ( ( x e. { A } /\ y e. { A } ) -> x = y )
5	4	3mix2d 1237	. . . 4 |- ( ( x e. { A } /\ y e. { A } ) -> ( x R y \/ x = y \/ y R x ) )
6	5	rgen2a 2977	. . 3 |- A. x e. { A } A. y e. { A } ( x R y \/ x = y \/ y R x )
7		df-so 5036	. . 3 |- ( R Or { A } <-> ( R Po { A } /\ A. x e. { A } A. y e. { A } ( x R y \/ x = y \/ y R x ) ) )
8	6, 7	mpbiran2 954	. 2 |- ( R Or { A } <-> R Po { A } )
9		posn 5187	. 2 |- ( Rel R -> ( R Po { A } <-> -. A R A ) )
10	8, 9	syl5bb 272	1 |- ( Rel R -> ( R Or { A } <-> -. A R A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   \/ w3o 1036   e. wcel 1990   A.wral 2912   {csn 4177   class class class wbr 4653   Po wpo 5033   Or wor 5034   Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121

This theorem is referenced by:  wesn  5190