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

Theorem en2lp 7922
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 7921 . . 3  |-  _E  Fr  _V
2 efrn2lp 4802 . . 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 3079 . . . 4  |-  ( A  e.  B  ->  A  e.  _V )
5 elex 3079 . . . 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 1758   _Vcvv 3070    _E cep 4730    Fr wfr 4776
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pr 4631  ax-reg 7910
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-sbc 3287  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-sn 3978  df-pr 3980  df-op 3984  df-br 4393  df-opab 4451  df-eprel 4732  df-fr 4779
This theorem is referenced by:  preleq  7926  suc11reg  7928  axunndlem1  8862  axacndlem5  8881  tratrb  31544  tratrbVD  31899
  Copyright terms: Public domain W3C validator