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

Theorem unssi 3178
Description: An inference showing the union of two subclasses is a subclass. (Contributed by Raph Levien, 10-Dec-2002.)
Hypotheses
Ref Expression
unssi.1  |-  A  C_  C
unssi.2  |-  B  C_  C
Assertion
Ref Expression
unssi  |-  ( A  u.  B )  C_  C

Proof of Theorem unssi
StepHypRef Expression
1 unssi.1 . . 3  |-  A  C_  C
2 unssi.2 . . 3  |-  B  C_  C
31, 2pm3.2i 267 . 2  |-  ( A 
C_  C  /\  B  C_  C )
4 unss 3177 . 2  |-  ( ( A  C_  C  /\  B  C_  C )  <->  ( A  u.  B )  C_  C
)
53, 4mpbi 144 1  |-  ( A  u.  B )  C_  C
Colors of variables: wff set class
Syntax hints:    /\ wa 103    u. cun 3000    C_ wss 3002
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-un 3006  df-in 3008  df-ss 3015
This theorem is referenced by:  undifabs  3365  inundifss  3366  dmrnssfld  4711  djuun  6816  caserel  6834  ltrelxr  7610  nn0ssre  8740  nn0ssz  8831  strleun  11646
  Copyright terms: Public domain W3C validator