Theorem efrirr 4384

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		frirrg 4381	. . . 4 |- ( ( _E Fr A /\ A e. A /\ A e. A ) -> -. A _E A )
2	1	3anidm23 1308	. . 3 |- ( ( _E Fr A /\ A e. A ) -> -. A _E A )
3		epelg 4321	. . . 4 |- ( A e. A -> ( A _E A <-> A e. A ) )
4	3	adantl 277	. . 3 |- ( ( _E Fr A /\ A e. A ) -> ( A _E A <-> A e. A ) )
5	2, 4	mtbid 673	. 2 |- ( ( _E Fr A /\ A e. A ) -> -. A e. A )
6	5	pm2.01da 637	1 |- ( _E Fr A -> -. A e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2164   class class class wbr 4029   _E cep 4318   Fr wfr 4359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-eprel 4320  df-frfor 4362  df-frind 4363

This theorem is referenced by:  tz7.2  4385