Theorem bj-ru 32934

Description: Remove dependency on ax-13 2246 (and df-v 3202) from Russell's paradox ru 3434 expressed with primitive symbols and with a class variable V (note that axsep2 4782 does require ax-8 1992 and ax-9 1999 since it requires df-clel 2618 and df-cleq 2615--- see bj-df-clel 32888 and bj-df-cleq 32893). Note the more economical use of bj-elissetv 32861 instead of isset 3207 to avoid use of df-v 3202. (Contributed by BJ, 12-Oct-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-ru	|- -. { x | -. x e. x } e. V

Proof of Theorem bj-ru

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-ru1 32933	. 2 |- -. E. y y = { x | -. x e. x }
2		bj-elissetv 32861	. 2 |- ( { x | -. x e. x } e. V -> E. y y = { x | -. x e. x } )
3	1, 2	mto 188	1 |- -. { x | -. x e. x } e. V

Syntax hints:   -. wn 3   = wceq 1483   E.wex 1704   e. wcel 1990   {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: (None)