ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  undif2ss Unicode version

Theorem undif2ss 3408
Description: Absorption of difference by union. In classical logic, as in Part of proof of Corollary 6K of [Enderton] p. 144, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
Assertion
Ref Expression
undif2ss  |-  ( A  u.  ( B  \  A ) )  C_  ( A  u.  B
)

Proof of Theorem undif2ss
StepHypRef Expression
1 undif1ss 3407 . 2  |-  ( ( B  \  A )  u.  A )  C_  ( B  u.  A
)
2 uncom 3190 . 2  |-  ( A  u.  ( B  \  A ) )  =  ( ( B  \  A )  u.  A
)
3 uncom 3190 . 2  |-  ( A  u.  B )  =  ( B  u.  A
)
41, 2, 33sstr4i 3108 1  |-  ( A  u.  ( B  \  A ) )  C_  ( A  u.  B
)
Colors of variables: wff set class
Syntax hints:    \ cdif 3038    u. cun 3039    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-un 3045  df-in 3047  df-ss 3054
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator