Theorem en3lp 8513

Description: No class has 3-cycle membership loops. This proof was automatically generated from the virtual deduction proof en3lpVD 39080 using a translation program. (Contributed by Alan Sare, 24-Oct-2011.)

Assertion

Ref	Expression
en3lp	|- -. ( A e. B /\ B e. C /\ C e. A )

Proof of Theorem en3lp

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3919	. . . . 5 |- -. C e. (/)
2		eleq2 2690	. . . . 5 |- ( { A , B , C } = (/) -> ( C e. { A , B , C } <-> C e. (/) ) )
3	1, 2	mtbiri 317	. . . 4 |- ( { A , B , C } = (/) -> -. C e. { A , B , C } )
4		tpid3g 4305	. . . 4 |- ( C e. A -> C e. { A , B , C } )
5	3, 4	nsyl 135	. . 3 |- ( { A , B , C } = (/) -> -. C e. A )
6	5	intn3an3d 1444	. 2 |- ( { A , B , C } = (/) -> -. ( A e. B /\ B e. C /\ C e. A ) )
7		tpex 6957	. . . 4 |- { A , B , C } e. _V
8		zfreg 8500	. . . 4 |- ( ( { A , B , C } e. _V /\ { A , B , C } =/= (/) ) -> E. x e. { A , B , C } ( x i^i { A , B , C } ) = (/) )
9	7, 8	mpan 706	. . 3 |- ( { A , B , C } =/= (/) -> E. x e. { A , B , C } ( x i^i { A , B , C } ) = (/) )
10		en3lplem2 8512	. . . . . 6 |- ( ( A e. B /\ B e. C /\ C e. A ) -> ( x e. { A , B , C } -> ( x i^i { A , B , C } ) =/= (/) ) )
11	10	com12 32	. . . . 5 |- ( x e. { A , B , C } -> ( ( A e. B /\ B e. C /\ C e. A ) -> ( x i^i { A , B , C } ) =/= (/) ) )
12	11	necon2bd 2810	. . . 4 |- ( x e. { A , B , C } -> ( ( x i^i { A , B , C } ) = (/) -> -. ( A e. B /\ B e. C /\ C e. A ) ) )
13	12	rexlimiv 3027	. . 3 |- ( E. x e. { A , B , C } ( x i^i { A , B , C } ) = (/) -> -. ( A e. B /\ B e. C /\ C e. A ) )
14	9, 13	syl 17	. 2 |- ( { A , B , C } =/= (/) -> -. ( A e. B /\ B e. C /\ C e. A ) )
15	6, 14	pm2.61ine 2877	1 |- -. ( A e. B /\ B e. C /\ C e. A )

Syntax hints:   -. wn 3   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   _Vcvv 3200   i^i cin 3573   (/)c0 3915   {ctp 4181

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-8 1992  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  ax-un 6949  ax-reg 8497

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-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-nul 3916  df-sn 4178  df-pr 4180  df-tp 4182  df-uni 4437

This theorem is referenced by:  bj-inftyexpidisj  33097  tratrb  38746  tratrbVD  39097