Theorem npss 3717

Description: A class is not a proper subclass of another iff it satisfies a one-directional form of eqss 3618. (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		pm4.61 442	. . 3 |- ( -. ( A C_ B -> A = B ) <-> ( A C_ B /\ -. A = B ) )
2		dfpss2 3692	. . 3 |- ( A C. B <-> ( A C_ B /\ -. A = B ) )
3	1, 2	bitr4i 267	. 2 |- ( -. ( A C_ B -> A = B ) <-> A C. B )
4	3	con1bii 346	1 |- ( -. A C. B <-> ( A C_ B -> A = B ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   C_ wss 3574   C. wpss 3575

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 197  df-an 386  df-ne 2795  df-pss 3590

This theorem is referenced by:  ttukeylem7  9337  canthp1lem2  9475  pgpfac1lem1  18473  lspsncv0  19146  obslbs  20074