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Theorem uniss 3629
Description: Subclass relationship for class union. Theorem 61 of [Suppes] p. 39. (Contributed by NM, 22-Mar-1998.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
uniss  |-  ( A 
C_  B  ->  U. A  C_ 
U. B )

Proof of Theorem uniss
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 2967 . . . . 5  |-  ( A 
C_  B  ->  (
y  e.  A  -> 
y  e.  B ) )
21anim2d 324 . . . 4  |-  ( A 
C_  B  ->  (
( x  e.  y  /\  y  e.  A
)  ->  ( x  e.  y  /\  y  e.  B ) ) )
32eximdv 1776 . . 3  |-  ( A 
C_  B  ->  ( E. y ( x  e.  y  /\  y  e.  A )  ->  E. y
( x  e.  y  /\  y  e.  B
) ) )
4 eluni 3611 . . 3  |-  ( x  e.  U. A  <->  E. y
( x  e.  y  /\  y  e.  A
) )
5 eluni 3611 . . 3  |-  ( x  e.  U. B  <->  E. y
( x  e.  y  /\  y  e.  B
) )
63, 4, 53imtr4g 198 . 2  |-  ( A 
C_  B  ->  (
x  e.  U. A  ->  x  e.  U. B
) )
76ssrdv 2979 1  |-  ( A 
C_  B  ->  U. A  C_ 
U. B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101   E.wex 1397    e. wcel 1409    C_ wss 2945   U.cuni 3608
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-uni 3609
This theorem is referenced by:  unissi  3631  unissd  3632  intssuni2m  3667  relfld  4874
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