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Theorem intssuni 4261
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 3880 . . . 4  |-  ( ( A  =/=  (/)  /\  A. y  e.  A  x  e.  y )  ->  E. y  e.  A  x  e.  y )
21ex 434 . . 3  |-  ( A  =/=  (/)  ->  ( A. y  e.  A  x  e.  y  ->  E. y  e.  A  x  e.  y ) )
3 vex 3081 . . . 4  |-  x  e. 
_V
43elint2 4246 . . 3  |-  ( x  e.  |^| A  <->  A. y  e.  A  x  e.  y )
5 eluni2 4206 . . 3  |-  ( x  e.  U. A  <->  E. y  e.  A  x  e.  y )
62, 4, 53imtr4g 270 . 2  |-  ( A  =/=  (/)  ->  ( x  e.  |^| A  ->  x  e.  U. A ) )
76ssrdv 3473 1  |-  ( A  =/=  (/)  ->  |^| A  C_  U. A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1758    =/= wne 2648   A.wral 2799   E.wrex 2800    C_ wss 3439   (/)c0 3748   U.cuni 4202   |^|cint 4239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-v 3080  df-dif 3442  df-in 3446  df-ss 3453  df-nul 3749  df-uni 4203  df-int 4240
This theorem is referenced by:  unissint  4263  intssuni2  4264  fin23lem31  8626  wunint  8996  tskint  9066  incexc  13421  incexc2  13422  subgint  15827  efgval  16338  lbsextlem3  17367  cssmre  18246  uffixfr  19631  uffix2  19632  uffixsn  19633  insiga  26745  dfon2lem8  27767  intidl  28997  elrfi  29198
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