Theorem vss 1729

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 1520	. . . 4 |- A (_ V
2	1	jctl 238	. . 3 |- (V (_ A -> (A (_ V /\ V (_ A))
3		eqss 1516	. . 3 |- (A = V <-> (A (_ V /\ V (_ A))
4	2, 3	sylibr 175	. 2 |- (V (_ A -> A = V)
5		eqimss2 1549	. 2 |- (A = V -> V (_ A)
6	4, 5	impbi 139	1 |- (V (_ A <-> A = V)

Syntax hints:   <-> wb 127   /\ wa 196   = wceq 1091  Vcvv 1348   (_ wss 1487

This theorem is referenced by:  vdif0 1749  dmen 3310

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492

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