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Theorem intssuni 3884
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 3543 . . . 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 2791 . . . 4  |-  x  e. 
_V
43elint2 3869 . . 3  |-  ( x  e.  |^| A  <->  A. y  e.  A  x  e.  y )
5 eluni2 3831 . . 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 3185 1  |-  ( A  =/=  (/)  ->  |^| A  C_  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   (/)c0 3455   U.cuni 3827   |^|cint 3862
This theorem is referenced by:  unissint  3886  intssuni2  3887  fin23lem31  7969  wunint  8337  tskint  8407  incexc  12296  incexc2  12297  subgint  14641  efgval  15026  lbsextlem3  15913  cssmre  16593  uffixfr  17618  uffix2  17619  uffixsn  17620  insiga  23498  dfon2lem8  24146  inttrp  25108  intidl  26654  elrfi  26769
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-ne 2448  df-ral 2548  df-rex 2549  df-v 2790  df-dif 3155  df-in 3159  df-ss 3166  df-nul 3456  df-uni 3828  df-int 3863
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