MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  en2lp Structured version   Unicode version

Theorem en2lp 8021
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 8020 . . 3  |-  _E  Fr  _V
2 efrn2lp 4856 . . 3  |-  ( (  _E  Fr  _V  /\  ( A  e.  _V  /\  B  e.  _V )
)  ->  -.  ( A  e.  B  /\  B  e.  A )
)
31, 2mpan 670 . 2  |-  ( ( A  e.  _V  /\  B  e.  _V )  ->  -.  ( A  e.  B  /\  B  e.  A ) )
4 elex 3117 . . . 4  |-  ( A  e.  B  ->  A  e.  _V )
5 elex 3117 . . . 4  |-  ( B  e.  A  ->  B  e.  _V )
64, 5anim12i 566 . . 3  |-  ( ( A  e.  B  /\  B  e.  A )  ->  ( A  e.  _V  /\  B  e.  _V )
)
76con3i 135 . 2  |-  ( -.  ( A  e.  _V  /\  B  e.  _V )  ->  -.  ( A  e.  B  /\  B  e.  A ) )
83, 7pm2.61i 164 1  |-  -.  ( A  e.  B  /\  B  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    /\ wa 369    e. wcel 1762   _Vcvv 3108    _E cep 4784    Fr wfr 4830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681  ax-reg 8009
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-br 4443  df-opab 4501  df-eprel 4786  df-fr 4833
This theorem is referenced by:  preleq  8025  suc11reg  8027  axunndlem1  8961  axacndlem5  8980  tratrb  32263  tratrbVD  32618
  Copyright terms: Public domain W3C validator