Theorem bj-nul 33018

Description: Two formulations of the axiom of the empty set ax-nul 4789. Proposal: place it right before ax-nul 4789. (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 3207	. 2 |- ( (/) e. _V <-> E. x x = (/) )
2		eq0 3929	. . 3 |- ( x = (/) <-> A. y -. y e. x )
3	2	exbii 1774	. 2 |- ( E. x x = (/) <-> E. x A. y -. y e. x )
4	1, 3	bitri 264	1 |- ( (/) e. _V <-> E. x A. y -. y e. x )

Syntax hints:   -. wn 3   <-> wb 196   A.wal 1481   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)