Theorem bj-ru1 32933

Description: A version of Russell's paradox ru 3434 (see also bj-ru 32934). Note the more economical use of bj-abeq2 32773 instead of abeq2 2732 to avoid dependency on ax-13 2246. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ru1	|- -. E. y y = { x | -. x e. x }

Distinct variable group:   x, y

Proof of Theorem bj-ru1

Step	Hyp	Ref	Expression
1		bj-ru0 32932	. . 3 |- -. A. x ( x e. y <-> -. x e. x )
2		bj-abeq2 32773	. . 3 |- ( y = { x | -. x e. x } <-> A. x ( x e. y <-> -. x e. x ) )
3	1, 2	mtbir 313	. 2 |- -. y = { x | -. x e. x }
4	3	nex 1731	1 |- -. E. y y = { x | -. x e. x }

Syntax hints:   -. wn 3   <-> wb 196   A.wal 1481   = wceq 1483   E.wex 1704   {cab 2608

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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  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

This theorem is referenced by:  bj-ru  32934