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

Theorem en2lp 8136
 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

Proof of Theorem en2lp
StepHypRef Expression
1 zfregfr 8135 . . 3
2 efrn2lp 4821 . . 3
31, 2mpan 684 . 2
4 elex 3040 . . . 4
5 elex 3040 . . . 4
64, 5anim12i 576 . . 3
76con3i 142 . 2
83, 7pm2.61i 169 1
 Colors of variables: wff setvar class Syntax hints:   wn 3   wa 376   wcel 1904  cvv 3031   cep 4748   wfr 4795 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-reg 8125 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-eprel 4750  df-fr 4798 This theorem is referenced by:  preleq  8140  suc11reg  8142  axunndlem1  9038  axacndlem5  9054  tratrb  36967  tratrbVD  37321
 Copyright terms: Public domain W3C validator