Theorem ruv 4598

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 2774	. 2 |- _V = { x | x = x }
2		equid 1724	. . . 4 |- x = x
3		elirrv 4596	. . . . 5 |- -. x e. x
4	3	nelir 2474	. . . 4 |- x e/ x
5	2, 4	2th 174	. . 3 |- ( x = x <-> x e/ x )
6	5	abbii 2321	. 2 |- { x | x = x } = { x | x e/ x }
7	1, 6	eqtr2i 2227	1 |- { x | x e/ x } = _V

Syntax hints:   = wceq 1373   {cab 2191   e/ wnel 2471   _Vcvv 2772

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187  ax-setind 4585

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-v 2774  df-dif 3168  df-sn 3639

This theorem is referenced by:  ruALT  4599