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

Step	Hyp	Ref	Expression
1		isset 3035	. 2 |- ( (/) e. _V <-> E. x x = (/) )
2		eq0 3738	. . 3 |- ( x = (/) <-> A. y -. y e. x )
3	2	exbii 1726	. 2 |- ( E. x x = (/) <-> E. x A. y -. y e. x )
4	1, 3	bitri 257	1 |- ( (/) e. _V <-> E. x A. y -. y e. x )

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)