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Theorem efrirr 4245
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 frirrg 4242 . . . 4  |-  ( (  _E  Fr  A  /\  A  e.  A  /\  A  e.  A )  ->  -.  A  _E  A
)
213anidm23 1260 . . 3  |-  ( (  _E  Fr  A  /\  A  e.  A )  ->  -.  A  _E  A
)
3 epelg 4182 . . . 4  |-  ( A  e.  A  ->  ( A  _E  A  <->  A  e.  A ) )
43adantl 275 . . 3  |-  ( (  _E  Fr  A  /\  A  e.  A )  ->  ( A  _E  A  <->  A  e.  A ) )
52, 4mtbid 646 . 2  |-  ( (  _E  Fr  A  /\  A  e.  A )  ->  -.  A  e.  A
)
65pm2.01da 610 1  |-  (  _E  Fr  A  ->  -.  A  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    e. wcel 1465   class class class wbr 3899    _E cep 4179    Fr wfr 4220
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-eprel 4181  df-frfor 4223  df-frind 4224
This theorem is referenced by:  tz7.2  4246
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