Theorem efrirr 4860

Description: Irreflexivity of the epsilon relation: a class founded by epsilon is not a member of itself. (Contributed by NM, 18-Apr-1994.) (Revised by Mario Carneiro, 22-Jun-2015.)

Assertion

Ref	Expression
efrirr	|- ( _E Fr A -> -. A e. A )

Proof of Theorem efrirr

Step	Hyp	Ref	Expression
1		frirr 4856	. . 3 |- ( ( _E Fr A /\ A e. A ) -> -. A _E A )
2		epelg 4792	. . . 4 |- ( A e. A -> ( A _E A <-> A e. A ) )
3	2	adantl 466	. . 3 |- ( ( _E Fr A /\ A e. A ) -> ( A _E A <-> A e. A ) )
4	1, 3	mtbid 300	. 2 |- ( ( _E Fr A /\ A e. A ) -> -. A e. A )
5	4	pm2.01da 442	1 |- ( _E Fr A -> -. A e. A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   e. wcel 1767   class class class wbr 4447   _E cep 4789   Fr wfr 4835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-eprel 4791  df-fr 4838

This theorem is referenced by:  tz7.2  4863  ordirr  4896