Theorem ruALT 8508

Description: Alternate proof of ru 3434, simplified using (indirectly) the Axiom of Regularity ax-reg 8497. (Contributed by Alan Sare, 4-Oct-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
ruALT	|- { x | x e/ x } e/ _V

Proof of Theorem ruALT

Step	Hyp	Ref	Expression
1		vprc 4796	. . 3 |- -. _V e. _V
2	1	nelir 2900	. 2 |- _V e/ _V
3		ruv 8507	. . 3 |- { x | x e/ x } = _V
4		neleq1 2902	. . 3 |- ( { x | x e/ x } = _V -> ( { x | x e/ x } e/ _V <-> _V e/ _V ) )
5	3, 4	ax-mp 5	. 2 |- ( { x | x e/ x } e/ _V <-> _V e/ _V )
6	2, 5	mpbir 221	1 |- { x | x e/ x } e/ _V

Syntax hints:   <-> wb 196   = wceq 1483   {cab 2608   e/ wnel 2897   _Vcvv 3200

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-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-reg 8497

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-nel 2898  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180

This theorem is referenced by: (None)