Theorem ruv 5704

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 2294	. 2 |- _V = {x | x = x}
2		equid 1484	. . . 4 |- x = x
3		elirrv 5700	. . . . 5 |- -. x e. x
4		df-nel 2020	. . . . 5 |- (x e/ x <-> -. x e. x)
5	3, 4	mpbir 207	. . . 4 |- x e/ x
6	2, 5	2th 786	. . 3 |- (x = x <-> x e/ x)
7	6	abbii 2006	. 2 |- {x | x = x} = {x | x e/ x}
8	1, 7	eqtr2i 1909	1 |- {x | x e/ x} = _V

Syntax hints:  -. wn 2   = wceq 1298   e. wcel 1300  {cab 1871   e/ wnel 2018  _Vcvv 2292

This theorem is referenced by:  ruALT 5705

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-reg 5695

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049

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