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Theorem uniss2 3799
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. See iunss2 3888 for a generalization to indexed unions. (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 3790 . . . . 5  |-  ( ( x  C_  y  /\  y  e.  B )  ->  x  C_  U. B )
21expcom 426 . . . 4  |-  ( y  e.  B  ->  (
x  C_  y  ->  x 
C_  U. B ) )
32rexlimiv 2632 . . 3  |-  ( E. y  e.  B  x 
C_  y  ->  x  C_ 
U. B )
43ralimi 2589 . 2  |-  ( A. x  e.  A  E. y  e.  B  x  C_  y  ->  A. x  e.  A  x  C_  U. B
)
5 unissb 3798 . 2  |-  ( U. A  C_  U. B  <->  A. x  e.  A  x  C_  U. B
)
64, 5sylibr 205 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 6    e. wcel 1621   A.wral 2516   E.wrex 2517    C_ wss 3094   U.cuni 3768
This theorem is referenced by:  unidif  3800  coflim  7820  unint2t  24850  intfmu2  24851
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ral 2520  df-rex 2521  df-v 2742  df-in 3101  df-ss 3108  df-uni 3769
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