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Theorem efrirr 4523
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 4519 . . 3  |-  ( (  _E  Fr  A  /\  A  e.  A )  ->  -.  A  _E  A
)
2 epelg 4455 . . . 4  |-  ( A  e.  A  ->  ( A  _E  A  <->  A  e.  A ) )
32adantl 453 . . 3  |-  ( (  _E  Fr  A  /\  A  e.  A )  ->  ( A  _E  A  <->  A  e.  A ) )
41, 3mtbid 292 . 2  |-  ( (  _E  Fr  A  /\  A  e.  A )  ->  -.  A  e.  A
)
54pm2.01da 430 1  |-  (  _E  Fr  A  ->  -.  A  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1721   class class class wbr 4172    _E cep 4452    Fr wfr 4498
This theorem is referenced by:  tz7.2  4526  ordirr  4559
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-eprel 4454  df-fr 4501
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