Theorem bj-ru 31583

Description: Remove dependency on ax-13 2101 (and df-v 3058) from Russell's paradox ru 3277 expressed with primitive symbols and with a class variable V (note that axsep2 4539 does require ax-8 1899 and ax-9 1906 since it requires df-clel 2457 and df-cleq 2454--- see bj-df-clel 31540 and bj-df-cleq 31544). Note the more economical use of bj-elissetv 31514 instead of isset 3060 to avoid use of df-v 3058. (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 31582	. 2 |- -. E. y y = { x | -. x e. x }
2		bj-elissetv 31514	. 2 |- ( { x | -. x e. x } e. V -> E. y y = { x | -. x e. x } )
3	1, 2	mto 181	1 |- -. { x | -. x e. x } e. V

Syntax hints:   -. wn 3   = wceq 1454   E.wex 1673   e. wcel 1897   {cab 2447

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-ext 2441

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457

This theorem is referenced by: (None)