Theorem dfss6 3593

Description: Alternate definition of subclass relationship. (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 3591	. . 3 |- ( A C_ B <-> A. x ( x e. A -> x e. B ) )
2		notnotb 304	. . 3 |- ( A. x ( x e. A -> x e. B ) <-> -. -. A. x ( x e. A -> x e. B ) )
3	1, 2	bitri 264	. 2 |- ( A C_ B <-> -. -. A. x ( x e. A -> x e. B ) )
4		exanali 1786	. 2 |- ( E. x ( x e. A /\ -. x e. B ) <-> -. A. x ( x e. A -> x e. B ) )
5	3, 4	xchbinxr 325	1 |- ( A C_ B <-> -. E. x ( x e. A /\ -. x e. B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   e. wcel 1990   C_ wss 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588

This theorem is referenced by: (None)