MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  difss2d Structured version   Unicode version

Theorem difss2d 3484
Description: If a class is contained in a difference, it is contained in the minuend. Deduction form of difss2 3483. (Contributed by David Moews, 1-May-2017.)
Hypothesis
Ref Expression
difss2d.1  |-  ( ph  ->  A  C_  ( B  \  C ) )
Assertion
Ref Expression
difss2d  |-  ( ph  ->  A  C_  B )

Proof of Theorem difss2d
StepHypRef Expression
1 difss2d.1 . 2  |-  ( ph  ->  A  C_  ( B  \  C ) )
2 difss2 3483 . 2  |-  ( A 
C_  ( B  \  C )  ->  A  C_  B )
31, 2syl 16 1  |-  ( ph  ->  A  C_  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \ cdif 3323    C_ wss 3326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-v 2972  df-dif 3329  df-in 3333  df-ss 3340
This theorem is referenced by:  oacomf1olem  7001  numacn  8217  ramub1lem1  14085  ramub1lem2  14086  mreexexlem2d  14581  mreexexlem3d  14582  mreexexlem4d  14583  mreexexd  14584  acsfiindd  15345  dpjidcl  16555  dpjidclOLD  16562  clsval2  18652  llycmpkgen2  19121  1stckgen  19125  alexsublem  19614  bcthlem3  20835  neibastop2lem  28578  eldioph2lem2  29096
  Copyright terms: Public domain W3C validator