Theorem efrirr 4338

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 4335	. . . 4 |- ( ( _E Fr A /\ A e. A /\ A e. A ) -> -. A _E A )
2	1	3anidm23 1292	. . 3 |- ( ( _E Fr A /\ A e. A ) -> -. A _E A )
3		epelg 4275	. . . 4 |- ( A e. A -> ( A _E A <-> A e. A ) )
4	3	adantl 275	. . 3 |- ( ( _E Fr A /\ A e. A ) -> ( A _E A <-> A e. A ) )
5	2, 4	mtbid 667	. 2 |- ( ( _E Fr A /\ A e. A ) -> -. A e. A )
6	5	pm2.01da 631	1 |- ( _E Fr A -> -. A e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   e. wcel 2141   class class class wbr 3989   _E cep 4272   Fr wfr 4313

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-eprel 4274  df-frfor 4316  df-frind 4317

This theorem is referenced by:  tz7.2  4339