Theorem ruv 4551

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 2741	. 2 |- _V = { x | x = x }
2		equid 1701	. . . 4 |- x = x
3		elirrv 4549	. . . . 5 |- -. x e. x
4	3	nelir 2445	. . . 4 |- x e/ x
5	2, 4	2th 174	. . 3 |- ( x = x <-> x e/ x )
6	5	abbii 2293	. 2 |- { x | x = x } = { x | x e/ x }
7	1, 6	eqtr2i 2199	1 |- { x | x e/ x } = _V

Syntax hints:   = wceq 1353   {cab 2163   e/ wnel 2442   _Vcvv 2739

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 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159  ax-setind 4538

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-v 2741  df-dif 3133  df-sn 3600

This theorem is referenced by:  ruALT  4552