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Theorem en2lp 7201
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 7200 . . 3  |-  _E  Fr  _V
2 efrn2lp 4268 . . 3  |-  ( (  _E  Fr  _V  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  -.  ( A  e.  B  /\  B  e.  A )
)
31, 2mpan 654 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  -.  ( A  e.  B  /\  B  e.  A ) )
4 elex 2735 . . . 4  |-  ( A  e.  B  ->  A  e.  _V )
5 elex 2735 . . . 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 1621   _Vcvv 2727    _E cep 4196    Fr wfr 4242
This theorem is referenced by:  preleq  7202  suc11reg  7204  axunndlem1  8097  axacndlem5  8113  tratrb  26992  tratrbVD  27327
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-reg 7190
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-br 3921  df-opab 3975  df-eprel 4198  df-fr 4245
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