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Theorem intssuni 3900
Description: The intersection of a nonempty set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
Assertion
Ref Expression
intssuni  |-  ( A  =/=  (/)  ->  |^| A  C_  U. A )

Proof of Theorem intssuni
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r19.2z 3556 . . . 4  |-  ( ( A  =/=  (/)  /\  A. y  e.  A  x  e.  y )  ->  E. y  e.  A  x  e.  y )
21ex 423 . . 3  |-  ( A  =/=  (/)  ->  ( A. y  e.  A  x  e.  y  ->  E. y  e.  A  x  e.  y ) )
3 vex 2804 . . . 4  |-  x  e. 
_V
43elint2 3885 . . 3  |-  ( x  e.  |^| A  <->  A. y  e.  A  x  e.  y )
5 eluni2 3847 . . 3  |-  ( x  e.  U. A  <->  E. y  e.  A  x  e.  y )
62, 4, 53imtr4g 261 . 2  |-  ( A  =/=  (/)  ->  ( x  e.  |^| A  ->  x  e.  U. A ) )
76ssrdv 3198 1  |-  ( A  =/=  (/)  ->  |^| A  C_  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696    =/= wne 2459   A.wral 2556   E.wrex 2557    C_ wss 3165   (/)c0 3468   U.cuni 3843   |^|cint 3878
This theorem is referenced by:  unissint  3902  intssuni2  3903  fin23lem31  7985  wunint  8353  tskint  8423  incexc  12312  incexc2  12313  subgint  14657  efgval  15042  lbsextlem3  15929  cssmre  16609  uffixfr  17634  uffix2  17635  uffixsn  17636  insiga  23513  dfon2lem8  24217  inttrp  25211  intidl  26757  elrfi  26872
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469  df-uni 3844  df-int 3879
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