Theorem ruv 4616

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 2778	. 2 |- _V = { x | x = x }
2		equid 1725	. . . 4 |- x = x
3		elirrv 4614	. . . . 5 |- -. x e. x
4	3	nelir 2476	. . . 4 |- x e/ x
5	2, 4	2th 174	. . 3 |- ( x = x <-> x e/ x )
6	5	abbii 2323	. 2 |- { x | x = x } = { x | x e/ x }
7	1, 6	eqtr2i 2229	1 |- { x | x e/ x } = _V

Syntax hints:   = wceq 1373   {cab 2193   e/ wnel 2473   _Vcvv 2776

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-v 2778  df-dif 3176  df-sn 3649

This theorem is referenced by:  ruALT  4617