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Definition df-pss 2961
Description: Define proper subclass relationship between two classes. Definition 5.9 of [TakeutiZaring] p. 17. Note that  -.  A  C.  A (proved in pssirr 3072). Contrast this relationship with the relationship  A  C_  B (as defined in df-ss 2959). Other possible definitions are given by dfpss2 3057 and dfpss3 3058. (Contributed by NM, 7-Feb-1996.)
Assertion
Ref Expression
df-pss  |-  ( A 
C.  B  <->  ( A  C_  B  /\  A  =/= 
B ) )

Detailed syntax breakdown of Definition df-pss
StepHypRef Expression
1 cA . . 3  class  A
2 cB . . 3  class  B
31, 2wpss 2946 . 2  wff  A  C.  B
41, 2wss 2945 . . 3  wff  A  C_  B
51, 2wne 2220 . . 3  wff  A  =/= 
B
64, 5wa 101 . 2  wff  ( A 
C_  B  /\  A  =/=  B )
73, 6wb 102 1  wff  ( A 
C.  B  <->  ( A  C_  B  /\  A  =/= 
B ) )
Colors of variables: wff set class
This definition is referenced by:  dfpss2  3057  psseq1  3059  psseq2  3060  pssss  3067  pssne  3068  nssinpss  3197  0pss  3293  difsnpssim  3535
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