Theorem ruv 4744

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 1858	. 2 |- V = {x | x = x}
2		equid 1162	. . . 4 |- x = x
3		elirrv 4741	. . . . 5 |- -. x e. x
4		df-nel 1631	. . . . 5 |- (x e/ x <-> -. x e. x)
5	3, 4	mpbir 188	. . . 4 |- x e/ x
6	2, 5	2th 723	. . 3 |- (x = x <-> x e/ x)
7	6	abbii 1618	. 2 |- {x | x = x} = {x | x e/ x}
8	1, 7	eqtr2i 1539	1 |- {x | x e/ x} = V

Syntax hints:  -. wn 2   = wceq 992   e. wcel 994  {cab 1505   e/ wnel 1629  Vcvv 1857

This theorem is referenced by:  ruALT 4745

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-sep 2777  ax-pow 2818  ax-reg 4736

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-nel 1631  df-ral 1695  df-rex 1696  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471

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