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Theorem uniss2 4038
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 4128 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 4029 . . . . 5  |-  ( ( x  C_  y  /\  y  e.  B )  ->  x  C_  U. B )
21expcom 425 . . . 4  |-  ( y  e.  B  ->  (
x  C_  y  ->  x 
C_  U. B ) )
32rexlimiv 2816 . . 3  |-  ( E. y  e.  B  x 
C_  y  ->  x  C_ 
U. B )
43ralimi 2773 . 2  |-  ( A. x  e.  A  E. y  e.  B  x  C_  y  ->  A. x  e.  A  x  C_  U. B
)
5 unissb 4037 . 2  |-  ( U. A  C_  U. B  <->  A. x  e.  A  x  C_  U. B
)
64, 5sylibr 204 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 1725   A.wral 2697   E.wrex 2698    C_ wss 3312   U.cuni 4007
This theorem is referenced by:  unidif  4039  coflim  8130
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-in 3319  df-ss 3326  df-uni 4008
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