Theorem ruv 4473

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 2691	. 2 |- _V = { x | x = x }
2		equid 1678	. . . 4 |- x = x
3		elirrv 4471	. . . . 5 |- -. x e. x
4	3	nelir 2407	. . . 4 |- x e/ x
5	2, 4	2th 173	. . 3 |- ( x = x <-> x e/ x )
6	5	abbii 2256	. 2 |- { x | x = x } = { x | x e/ x }
7	1, 6	eqtr2i 2162	1 |- { x | x e/ x } = _V

Syntax hints:   = wceq 1332   {cab 2126   e/ wnel 2404   _Vcvv 2689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-setind 4460

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-v 2691  df-dif 3078  df-sn 3538

This theorem is referenced by:  ruALT  4474