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

Theorem zfcndreg 8898
Description: Axiom of Regularity ax-reg 7921, reproved from conditionless ZFC axioms. (Contributed by NM, 15-Aug-2003.) (Proof modification is discouraged.)
Assertion
Ref Expression
zfcndreg  |-  ( E. y  y  e.  x  ->  E. y ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x ) ) )
Distinct variable group:    x, y, z

Proof of Theorem zfcndreg
StepHypRef Expression
1 nfe1 1780 . 2  |-  F/ y E. y ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x ) )
2 axregnd 8884 . 2  |-  ( y  e.  x  ->  E. y
( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x )
) )
31, 2exlimi 1850 1  |-  ( E. y  y  e.  x  ->  E. y ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369   A.wal 1368   E.wex 1587    e. wcel 1758
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-reg 7921
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-v 3080  df-dif 3442  df-un 3444  df-nul 3749  df-sn 3989  df-pr 3991
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator