Theorem npss 3380

Description: A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3288. (Contributed by Mario Carneiro, 15-May-2015.)

Assertion

Ref	Expression
npss	|- (-. A C. B <-> (A C_ B -> A = B))

Proof of Theorem npss

Step	Hyp	Ref	Expression
1		pm4.61 415	. . 3 |- (-. (A C_ B -> A = B) <-> (A C_ B /\ -. A = B))
2		dfpss2 3355	. . 3 |- (A C. B <-> (A C_ B /\ -. A = B))
3	1, 2	bitr4i 243	. 2 |- (-. (A C_ B -> A = B) <-> A C. B)
4	3	con1bii 321	1 |- (-. A C. B <-> (A C_ B -> A = B))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   C_ wss 3258   C. wpss 3259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-an 360  df-ne 2519  df-pss 3262

This theorem is referenced by: (None)