Theorem frsn 5189

Description: Founded relation on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.) (Revised by Mario Carneiro, 23-Apr-2015.)

Assertion

Ref	Expression
frsn	|- ( Rel R -> ( R Fr { A } <-> -. A R A ) )

Proof of Theorem frsn

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

Step	Hyp	Ref	Expression
1		snprc 4253	. . . . . 6 |- ( -. A e. _V <-> { A } = (/) )
2		fr0 5093	. . . . . . 7 |- R Fr (/)
3		freq2 5085	. . . . . . 7 |- ( { A } = (/) -> ( R Fr { A } <-> R Fr (/) ) )
4	2, 3	mpbiri 248	. . . . . 6 |- ( { A } = (/) -> R Fr { A } )
5	1, 4	sylbi 207	. . . . 5 |- ( -. A e. _V -> R Fr { A } )
6	5	adantl 482	. . . 4 |- ( ( Rel R /\ -. A e. _V ) -> R Fr { A } )
7		brrelex 5156	. . . . 5 |- ( ( Rel R /\ A R A ) -> A e. _V )
8	7	stoic1a 1697	. . . 4 |- ( ( Rel R /\ -. A e. _V ) -> -. A R A )
9	6, 8	2thd 255	. . 3 |- ( ( Rel R /\ -. A e. _V ) -> ( R Fr { A } <-> -. A R A ) )
10	9	ex 450	. 2 |- ( Rel R -> ( -. A e. _V -> ( R Fr { A } <-> -. A R A ) ) )
11		df-fr 5073	. . . 4 |- ( R Fr { A } <-> A. x ( ( x C_ { A } /\ x =/= (/) ) -> E. y e. x A. z e. x -. z R y ) )
12		sssn 4358	. . . . . . . . . . 11 |- ( x C_ { A } <-> ( x = (/) \/ x = { A } ) )
13		neor 2885	. . . . . . . . . . 11 |- ( ( x = (/) \/ x = { A } ) <-> ( x =/= (/) -> x = { A } ) )
14	12, 13	sylbb 209	. . . . . . . . . 10 |- ( x C_ { A } -> ( x =/= (/) -> x = { A } ) )
15	14	imp 445	. . . . . . . . 9 |- ( ( x C_ { A } /\ x =/= (/) ) -> x = { A } )
16	15	adantl 482	. . . . . . . 8 |- ( ( A e. _V /\ ( x C_ { A } /\ x =/= (/) ) ) -> x = { A } )
17		eqimss 3657	. . . . . . . . . 10 |- ( x = { A } -> x C_ { A } )
18	17	adantl 482	. . . . . . . . 9 |- ( ( A e. _V /\ x = { A } ) -> x C_ { A } )
19		snnzg 4308	. . . . . . . . . . 11 |- ( A e. _V -> { A } =/= (/) )
20		neeq1 2856	. . . . . . . . . . 11 |- ( x = { A } -> ( x =/= (/) <-> { A } =/= (/) ) )
21	19, 20	syl5ibrcom 237	. . . . . . . . . 10 |- ( A e. _V -> ( x = { A } -> x =/= (/) ) )
22	21	imp 445	. . . . . . . . 9 |- ( ( A e. _V /\ x = { A } ) -> x =/= (/) )
23	18, 22	jca 554	. . . . . . . 8 |- ( ( A e. _V /\ x = { A } ) -> ( x C_ { A } /\ x =/= (/) ) )
24	16, 23	impbida 877	. . . . . . 7 |- ( A e. _V -> ( ( x C_ { A } /\ x =/= (/) ) <-> x = { A } ) )
25	24	imbi1d 331	. . . . . 6 |- ( A e. _V -> ( ( ( x C_ { A } /\ x =/= (/) ) -> E. y e. x A. z e. x -. z R y ) <-> ( x = { A } -> E. y e. x A. z e. x -. z R y ) ) )
26	25	albidv 1849	. . . . 5 |- ( A e. _V -> ( A. x ( ( x C_ { A } /\ x =/= (/) ) -> E. y e. x A. z e. x -. z R y ) <-> A. x ( x = { A } -> E. y e. x A. z e. x -. z R y ) ) )
27		snex 4908	. . . . . 6 |- { A } e. _V
28		raleq 3138	. . . . . . 7 |- ( x = { A } -> ( A. z e. x -. z R y <-> A. z e. { A } -. z R y ) )
29	28	rexeqbi1dv 3147	. . . . . 6 |- ( x = { A } -> ( E. y e. x A. z e. x -. z R y <-> E. y e. { A } A. z e. { A } -. z R y ) )
30	27, 29	ceqsalv 3233	. . . . 5 |- ( A. x ( x = { A } -> E. y e. x A. z e. x -. z R y ) <-> E. y e. { A } A. z e. { A } -. z R y )
31	26, 30	syl6bb 276	. . . 4 |- ( A e. _V -> ( A. x ( ( x C_ { A } /\ x =/= (/) ) -> E. y e. x A. z e. x -. z R y ) <-> E. y e. { A } A. z e. { A } -. z R y ) )
32	11, 31	syl5bb 272	. . 3 |- ( A e. _V -> ( R Fr { A } <-> E. y e. { A } A. z e. { A } -. z R y ) )
33		breq2 4657	. . . . . 6 |- ( y = A -> ( z R y <-> z R A ) )
34	33	notbid 308	. . . . 5 |- ( y = A -> ( -. z R y <-> -. z R A ) )
35	34	ralbidv 2986	. . . 4 |- ( y = A -> ( A. z e. { A } -. z R y <-> A. z e. { A } -. z R A ) )
36	35	rexsng 4219	. . 3 |- ( A e. _V -> ( E. y e. { A } A. z e. { A } -. z R y <-> A. z e. { A } -. z R A ) )
37		breq1 4656	. . . . 5 |- ( z = A -> ( z R A <-> A R A ) )
38	37	notbid 308	. . . 4 |- ( z = A -> ( -. z R A <-> -. A R A ) )
39	38	ralsng 4218	. . 3 |- ( A e. _V -> ( A. z e. { A } -. z R A <-> -. A R A ) )
40	32, 36, 39	3bitrd 294	. 2 |- ( A e. _V -> ( R Fr { A } <-> -. A R A ) )
41	10, 40	pm2.61d2 172	1 |- ( Rel R -> ( R Fr { A } <-> -. A R A ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   (/)c0 3915   {csn 4177   class class class wbr 4653   Fr wfr 5070   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-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-ne 2795  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-fr 5073  df-xp 5120  df-rel 5121

This theorem is referenced by:  wesn  5190