Theorem efrirr 4815

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 4811	. . 3 |- ( ( _E Fr A /\ A e. A ) -> -. A _E A )
2		epelg 4746	. . . 4 |- ( A e. A -> ( A _E A <-> A e. A ) )
3	2	adantl 468	. . 3 |- ( ( _E Fr A /\ A e. A ) -> ( A _E A <-> A e. A ) )
4	1, 3	mtbid 302	. 2 |- ( ( _E Fr A /\ A e. A ) -> -. A e. A )
5	4	pm2.01da 444	1 |- ( _E Fr A -> -. A e. A )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   e. wcel 1887   class class class wbr 4402   _E cep 4743   Fr wfr 4790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-eprel 4745  df-fr 4793

This theorem is referenced by:  tz7.2  4818  ordirr  5441