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Theorem zfcndreg 9049
Description: Axiom of Regularity ax-reg 8116, 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 1894 . 2  |-  F/ y E. y ( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x ) )
2 axregnd 9036 . 2  |-  ( y  e.  x  ->  E. y
( y  e.  x  /\  A. z ( z  e.  y  ->  -.  z  e.  x )
) )
31, 2exlimi 1972 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 370   A.wal 1435   E.wex 1657    e. wcel 1872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-reg 8116
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-v 3082  df-dif 3439  df-un 3441  df-nul 3762  df-sn 3999  df-pr 4001
This theorem is referenced by: (None)
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