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Theorem uniss2 3858
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 3947 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 3849 . . . . 5  |-  ( ( x  C_  y  /\  y  e.  B )  ->  x  C_  U. B )
21expcom 424 . . . 4  |-  ( y  e.  B  ->  (
x  C_  y  ->  x 
C_  U. B ) )
32rexlimiv 2661 . . 3  |-  ( E. y  e.  B  x 
C_  y  ->  x  C_ 
U. B )
43ralimi 2618 . 2  |-  ( A. x  e.  A  E. y  e.  B  x  C_  y  ->  A. x  e.  A  x  C_  U. B
)
5 unissb 3857 . 2  |-  ( U. A  C_  U. B  <->  A. x  e.  A  x  C_  U. B
)
64, 5sylibr 203 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 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   U.cuni 3827
This theorem is referenced by:  unidif  3859  coflim  7887  unint2t  25518  intfmu2  25519
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-in 3159  df-ss 3166  df-uni 3828
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