Theorem vss 2307

Description: Only the universal class has the universal class as a subclass.

Assertion

Ref	Expression
vss	|- (V (_ A <-> A = V)

Proof of Theorem vss

Step	Hyp	Ref	Expression
1		ssv 2081	. . . 4 |- A (_ V
2	1	jctl 290	. . 3 |- (V (_ A -> (A (_ V /\ V (_ A))
3		eqss 2077	. . 3 |- (A = V <-> (A (_ V /\ V (_ A))
4	2, 3	sylibr 200	. 2 |- (V (_ A -> A = V)
5		eqimss2 2110	. 2 |- (A = V -> V (_ A)
6	4, 5	impbi 157	1 |- (V (_ A <-> A = V)

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 956  Vcvv 1811   (_ wss 2047

This theorem is referenced by:  vdif0 2328  dmen 4407

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-in 2051  df-ss 2053

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