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

Theorem ssuni 3753
Description: Subclass relationship for class union. (Contributed by NM, 24-May-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
ssuni  |-  ( ( A  C_  B  /\  B  e.  C )  ->  A  C_  U. C )

Proof of Theorem ssuni
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq2 2201 . . . . . . 7  |-  ( x  =  B  ->  (
y  e.  x  <->  y  e.  B ) )
21imbi1d 230 . . . . . 6  |-  ( x  =  B  ->  (
( y  e.  x  ->  y  e.  U. C
)  <->  ( y  e.  B  ->  y  e.  U. C ) ) )
3 elunii 3736 . . . . . . 7  |-  ( ( y  e.  x  /\  x  e.  C )  ->  y  e.  U. C
)
43expcom 115 . . . . . 6  |-  ( x  e.  C  ->  (
y  e.  x  -> 
y  e.  U. C
) )
52, 4vtoclga 2747 . . . . 5  |-  ( B  e.  C  ->  (
y  e.  B  -> 
y  e.  U. C
) )
65imim2d 54 . . . 4  |-  ( B  e.  C  ->  (
( y  e.  A  ->  y  e.  B )  ->  ( y  e.  A  ->  y  e.  U. C ) ) )
76alimdv 1851 . . 3  |-  ( B  e.  C  ->  ( A. y ( y  e.  A  ->  y  e.  B )  ->  A. y
( y  e.  A  ->  y  e.  U. C
) ) )
8 dfss2 3081 . . 3  |-  ( A 
C_  B  <->  A. y
( y  e.  A  ->  y  e.  B ) )
9 dfss2 3081 . . 3  |-  ( A 
C_  U. C  <->  A. y
( y  e.  A  ->  y  e.  U. C
) )
107, 8, 93imtr4g 204 . 2  |-  ( B  e.  C  ->  ( A  C_  B  ->  A  C_ 
U. C ) )
1110impcom 124 1  |-  ( ( A  C_  B  /\  B  e.  C )  ->  A  C_  U. C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1329    = wceq 1331    e. wcel 1480    C_ wss 3066   U.cuni 3731
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-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-uni 3732
This theorem is referenced by:  elssuni  3759  uniss2  3762  ssorduni  4398
  Copyright terms: Public domain W3C validator