Theorem efrn2lp 5096

Description: A set founded by epsilon contains no 2-cycle loops. (Contributed by NM, 19-Apr-1994.)

Assertion

Ref	Expression
efrn2lp	|- ( ( _E Fr A /\ ( B e. A /\ C e. A ) ) -> -. ( B e. C /\ C e. B ) )

Proof of Theorem efrn2lp

Step	Hyp	Ref	Expression
1		fr2nr 5092	. 2 |- ( ( _E Fr A /\ ( B e. A /\ C e. A ) ) -> -. ( B _E C /\ C _E B ) )
2		epelg 5030	. . . 4 |- ( C e. A -> ( B _E C <-> B e. C ) )
3		epelg 5030	. . . 4 |- ( B e. A -> ( C _E B <-> C e. B ) )
4	2, 3	bi2anan9r 918	. . 3 |- ( ( B e. A /\ C e. A ) -> ( ( B _E C /\ C _E B ) <-> ( B e. C /\ C e. B ) ) )
5	4	adantl 482	. 2 |- ( ( _E Fr A /\ ( B e. A /\ C e. A ) ) -> ( ( B _E C /\ C _E B ) <-> ( B e. C /\ C e. B ) ) )
6	1, 5	mtbid 314	1 |- ( ( _E Fr A /\ ( B e. A /\ C e. A ) ) -> -. ( B e. C /\ C e. B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   e. wcel 1990   class class class wbr 4653   _E cep 5028   Fr wfr 5070

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-eu 2474  df-mo 2475  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-eprel 5029  df-fr 5073

This theorem is referenced by:  en2lp  8510