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Theorem bj-nul 31692
Description: Two formulations of the axiom of the empty set ax-nul 4527. Proposal: place it right before ax-nul 4527. (Contributed by BJ, 30-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-nul  |-  ( (/)  e.  _V  <->  E. x A. y  -.  y  e.  x
)
Distinct variable group:    x, y

Proof of Theorem bj-nul
StepHypRef Expression
1 isset 3035 . 2  |-  ( (/)  e.  _V  <->  E. x  x  =  (/) )
2 eq0 3738 . . 3  |-  ( x  =  (/)  <->  A. y  -.  y  e.  x )
32exbii 1726 . 2  |-  ( E. x  x  =  (/)  <->  E. x A. y  -.  y  e.  x )
41, 3bitri 257 1  |-  ( (/)  e.  _V  <->  E. x A. y  -.  y  e.  x
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 189   A.wal 1450    = wceq 1452   E.wex 1671    e. wcel 1904   _Vcvv 3031   (/)c0 3722
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-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-v 3033  df-dif 3393  df-nul 3723
This theorem is referenced by: (None)
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