Theorem ruv 4543

Description: The Russell class is equal to the universe _V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)

Assertion

Ref	Expression
ruv	|- { x | x e/ x } = _V

Proof of Theorem ruv

Step	Hyp	Ref	Expression
1		df-v 2737	. 2 |- _V = { x | x = x }
2		equid 1699	. . . 4 |- x = x
3		elirrv 4541	. . . . 5 |- -. x e. x
4	3	nelir 2443	. . . 4 |- x e/ x
5	2, 4	2th 174	. . 3 |- ( x = x <-> x e/ x )
6	5	abbii 2291	. 2 |- { x | x = x } = { x | x e/ x }
7	1, 6	eqtr2i 2197	1 |- { x | x e/ x } = _V

Syntax hints:   = wceq 1353   {cab 2161   e/ wnel 2440   _Vcvv 2735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157  ax-setind 4530

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-v 2737  df-dif 3129  df-sn 3595

This theorem is referenced by:  ruALT  4544