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Theorem efrirr 4835
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
StepHypRef Expression
1 frirr 4831 . . 3  |-  ( (  _E  Fr  A  /\  A  e.  A )  ->  -.  A  _E  A
)
2 epelg 4766 . . . 4  |-  ( A  e.  A  ->  ( A  _E  A  <->  A  e.  A ) )
32adantl 467 . . 3  |-  ( (  _E  Fr  A  /\  A  e.  A )  ->  ( A  _E  A  <->  A  e.  A ) )
41, 3mtbid 301 . 2  |-  ( (  _E  Fr  A  /\  A  e.  A )  ->  -.  A  e.  A
)
54pm2.01da 443 1  |-  (  _E  Fr  A  ->  -.  A  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    e. wcel 1870   class class class wbr 4426    _E cep 4763    Fr wfr 4810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-eprel 4765  df-fr 4813
This theorem is referenced by:  tz7.2  4838  ordirr  5460
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