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Theorem en2lp 7312
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 7311 . . 3  |-  _E  Fr  _V
2 efrn2lp 4374 . . 3  |-  ( (  _E  Fr  _V  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  -.  ( A  e.  B  /\  B  e.  A )
)
31, 2mpan 653 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  -.  ( A  e.  B  /\  B  e.  A ) )
4 elex 2797 . . . 4  |-  ( A  e.  B  ->  A  e.  _V )
5 elex 2797 . . . 4  |-  ( B  e.  A  ->  B  e.  _V )
64, 5anim12i 551 . . 3  |-  ( ( A  e.  B  /\  B  e.  A )  ->  ( A  e.  _V  /\  B  e.  _V )
)
76con3i 129 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  -.  ( A  e.  B  /\  B  e.  A ) )
83, 7pm2.61i 158 1  |-  -.  ( A  e.  B  /\  B  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 5    /\ wa 360    e. wcel 1685   _Vcvv 2789    _E cep 4302    Fr wfr 4348
This theorem is referenced by:  preleq  7313  suc11reg  7315  axunndlem1  8212  axacndlem5  8228  tratrb  27570  tratrbVD  27905
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-reg 7301
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-eprel 4304  df-fr 4351
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