MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssneld Structured version   Unicode version

Theorem ssneld 3501
Description: If a class is not in another class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
ssneld.1  |-  ( ph  ->  A  C_  B )
Assertion
Ref Expression
ssneld  |-  ( ph  ->  ( -.  C  e.  B  ->  -.  C  e.  A ) )

Proof of Theorem ssneld
StepHypRef Expression
1 ssneld.1 . . 3  |-  ( ph  ->  A  C_  B )
21sseld 3498 . 2  |-  ( ph  ->  ( C  e.  A  ->  C  e.  B ) )
32con3d 133 1  |-  ( ph  ->  ( -.  C  e.  B  ->  -.  C  e.  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1762    C_ wss 3471
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-in 3478  df-ss 3485
This theorem is referenced by:  ssneldd  3502  kmlem2  8522  hashbclem  12456  mrissmrid  14887  mpfrcl  17953  prodss  28644  onsuct0  29471  ftc1anc  29664  dvhdimlem  36118  dvh3dim2  36122  dvh3dim3N  36123  mapdh9a  36464  hdmapval0  36510  hdmap11lem2  36519
  Copyright terms: Public domain W3C validator