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Theorem efrirr 4815
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 4811 . . 3  |-  ( (  _E  Fr  A  /\  A  e.  A )  ->  -.  A  _E  A
)
2 epelg 4746 . . . 4  |-  ( A  e.  A  ->  ( A  _E  A  <->  A  e.  A ) )
32adantl 468 . . 3  |-  ( (  _E  Fr  A  /\  A  e.  A )  ->  ( A  _E  A  <->  A  e.  A ) )
41, 3mtbid 302 . 2  |-  ( (  _E  Fr  A  /\  A  e.  A )  ->  -.  A  e.  A
)
54pm2.01da 444 1  |-  (  _E  Fr  A  ->  -.  A  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371    e. wcel 1887   class class class wbr 4402    _E cep 4743    Fr wfr 4790
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-br 4403  df-opab 4462  df-eprel 4745  df-fr 4793
This theorem is referenced by:  tz7.2  4818  ordirr  5441
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