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Theorem uniss2 3735
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 3726 . . . . 5  |-  ( ( x  C_  y  /\  y  e.  B )  ->  x  C_  U. B )
21expcom 115 . . . 4  |-  ( y  e.  B  ->  (
x  C_  y  ->  x 
C_  U. B ) )
32rexlimiv 2518 . . 3  |-  ( E. y  e.  B  x 
C_  y  ->  x  C_ 
U. B )
43ralimi 2470 . 2  |-  ( A. x  e.  A  E. y  e.  B  x  C_  y  ->  A. x  e.  A  x  C_  U. B
)
5 unissb 3734 . 2  |-  ( U. A  C_  U. B  <->  A. x  e.  A  x  C_  U. B
)
64, 5sylibr 133 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 1463   A.wral 2391   E.wrex 2392    C_ wss 3039   U.cuni 3704
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-in 3045  df-ss 3052  df-uni 3705
This theorem is referenced by:  unidif  3736
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