Theorem dfpss2 3029

Description: Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.)

Assertion

Ref	Expression
dfpss2	|- ( A C. B <-> ( A C_ B /\ -. A = B ) )

Proof of Theorem dfpss2

Step	Hyp	Ref	Expression
1		df-pss 2933	. 2 |- ( A C. B <-> ( A C_ B /\ A =/= B ) )
2		df-ne 2206	. . 3 |- ( A =/= B <-> -. A = B )
3	2	anbi2i 430	. 2 |- ( ( A C_ B /\ A =/= B ) <-> ( A C_ B /\ -. A = B ) )
4	1, 3	bitri 173	1 |- ( A C. B <-> ( A C_ B /\ -. A = B ) )

Syntax hints:   -. wn 3   /\ wa 97   <-> wb 98   = wceq 1243   =/= wne 2204   C_ wss 2917   C. wpss 2918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101

This theorem depends on definitions:  df-bi 110  df-ne 2206  df-pss 2933

This theorem is referenced by:  dfpss3  3030  psstr  3049  sspsstr  3050  psssstr  3051  pssv  3267  disj4im  3276  ssnelpss  3289  onpsssuc  4295  f1imapss  5415