Theorem efrirr 4400

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 4397	. . . 4 |- ( ( _E Fr A /\ A e. A /\ A e. A ) -> -. A _E A )
2	1	3anidm23 1310	. . 3 |- ( ( _E Fr A /\ A e. A ) -> -. A _E A )
3		epelg 4337	. . . 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 674	. 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 2176   class class class wbr 4044   _E cep 4334   Fr wfr 4375

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4045  df-opab 4106  df-eprel 4336  df-frfor 4378  df-frind 4379

This theorem is referenced by:  tz7.2  4401