Theorem efrirr 4136

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 4133	. . . 4 |- ( ( _E Fr A /\ A e. A /\ A e. A ) -> -. A _E A )
2	1	3anidm23 1229	. . 3 |- ( ( _E Fr A /\ A e. A ) -> -. A _E A )
3		epelg 4073	. . . 4 |- ( A e. A -> ( A _E A <-> A e. A ) )
4	3	adantl 271	. . 3 |- ( ( _E Fr A /\ A e. A ) -> ( A _E A <-> A e. A ) )
5	2, 4	mtbid 630	. 2 |- ( ( _E Fr A /\ A e. A ) -> -. A e. A )
6	5	pm2.01da 598	1 |- ( _E Fr A -> -. A e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   e. wcel 1434   class class class wbr 3805   _E cep 4070   Fr wfr 4111

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-v 2612  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-eprel 4072  df-frfor 4114  df-frind 4115

This theorem is referenced by:  tz7.2  4137