Theorem dfss6 36375

Description: Another definition of subclasshood. (Contributed by RP, 16-Apr-2020.)

Assertion

Ref	Expression
dfss6	|- ( A C_ B <-> -. E. x ( x e. A /\ -. x e. B ) )

Distinct variable groups:   x, A   x, B

Proof of Theorem dfss6

Step	Hyp	Ref	Expression
1		dfss2 3423	. . 3 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
2		notnot 293	. . 3 |- ( A. x ( x e. A -> x e. B ) <-> -. -. A. x ( x e. A -> x e. B ) )
3	1, 2	bitri 253	. 2 |- ( A C_ B <-> -. -. A. x ( x e. A -> x e. B ) )
4		exanali 1723	. 2 |- ( E. x ( x e. A /\ -. x e. B ) <-> -. A. x ( x e. A -> x e. B ) )
5	3, 4	xchbinxr 313	1 |- ( A C_ B <-> -. E. x ( x e. A /\ -. x e. B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   A.wal 1444   E.wex 1665   e. wcel 1889   C_ wss 3406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433

This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-in 3413  df-ss 3420

This theorem is referenced by: (None)