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Theorem en2lp 7317
Description: No class has 2-cycle membership loops. Theorem 7X(b) of [Enderton] p. 206. (Contributed by NM, 16-Oct-1996.) (Revised by Mario Carneiro, 25-Jun-2015.)
Assertion
Ref Expression
en2lp  |-  -.  ( A  e.  B  /\  B  e.  A )

Proof of Theorem en2lp
StepHypRef Expression
1 zfregfr 7316 . . 3  |-  _E  Fr  _V
2 efrn2lp 4375 . . 3  |-  ( (  _E  Fr  _V  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  -.  ( A  e.  B  /\  B  e.  A )
)
31, 2mpan 651 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  -.  ( A  e.  B  /\  B  e.  A ) )
4 elex 2796 . . . 4  |-  ( A  e.  B  ->  A  e.  _V )
5 elex 2796 . . . 4  |-  ( B  e.  A  ->  B  e.  _V )
64, 5anim12i 549 . . 3  |-  ( ( A  e.  B  /\  B  e.  A )  ->  ( A  e.  _V  /\  B  e.  _V )
)
76con3i 127 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  -.  ( A  e.  B  /\  B  e.  A ) )
83, 7pm2.61i 156 1  |-  -.  ( A  e.  B  /\  B  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358    e. wcel 1684   _Vcvv 2788    _E cep 4303    Fr wfr 4349
This theorem is referenced by:  preleq  7318  suc11reg  7320  axunndlem1  8217  axacndlem5  8233  tratrb  28299  tratrbVD  28637
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-reg 7306
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-eprel 4305  df-fr 4352
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