ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssdif2d Unicode version

Theorem ssdif2d 3185
Description: If  A is contained in  B and  C is contained in  D, then  ( A  \  D ) is contained in  ( B  \  C ). Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
ssdifd.1  |-  ( ph  ->  A  C_  B )
ssdif2d.2  |-  ( ph  ->  C  C_  D )
Assertion
Ref Expression
ssdif2d  |-  ( ph  ->  ( A  \  D
)  C_  ( B  \  C ) )

Proof of Theorem ssdif2d
StepHypRef Expression
1 ssdif2d.2 . . 3  |-  ( ph  ->  C  C_  D )
21sscond 3183 . 2  |-  ( ph  ->  ( A  \  D
)  C_  ( A  \  C ) )
3 ssdifd.1 . . 3  |-  ( ph  ->  A  C_  B )
43ssdifd 3182 . 2  |-  ( ph  ->  ( A  \  C
)  C_  ( B  \  C ) )
52, 4sstrd 3077 1  |-  ( ph  ->  ( A  \  D
)  C_  ( B  \  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \ cdif 3038    C_ wss 3041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-dif 3043  df-in 3047  df-ss 3054
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator