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Theorem uniss2 3640
Description: A subclass condition on the members of two classes that implies a subclass relation on their unions. Proposition 8.6 of [TakeutiZaring] p. 59. (Contributed by NM, 22-Mar-2004.)
Assertion
Ref Expression
uniss2  |-  ( A. x  e.  A  E. y  e.  B  x  C_  y  ->  U. A  C_  U. B )
Distinct variable groups:    x, A    x, y, B
Allowed substitution hint:    A( y)

Proof of Theorem uniss2
StepHypRef Expression
1 ssuni 3631 . . . . 5  |-  ( ( x  C_  y  /\  y  e.  B )  ->  x  C_  U. B )
21expcom 114 . . . 4  |-  ( y  e.  B  ->  (
x  C_  y  ->  x 
C_  U. B ) )
32rexlimiv 2472 . . 3  |-  ( E. y  e.  B  x 
C_  y  ->  x  C_ 
U. B )
43ralimi 2427 . 2  |-  ( A. x  e.  A  E. y  e.  B  x  C_  y  ->  A. x  e.  A  x  C_  U. B
)
5 unissb 3639 . 2  |-  ( U. A  C_  U. B  <->  A. x  e.  A  x  C_  U. B
)
64, 5sylibr 132 1  |-  ( A. x  e.  A  E. y  e.  B  x  C_  y  ->  U. A  C_  U. B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1434   A.wral 2349   E.wrex 2350    C_ wss 2974   U.cuni 3609
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-in 2980  df-ss 2987  df-uni 3610
This theorem is referenced by:  unidif  3641
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