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Theorem intssuni 4311
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 3921 . . . 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 3112 . . . 4  |-  x  e. 
_V
43elint2 4295 . . 3  |-  ( x  e.  |^| A  <->  A. y  e.  A  x  e.  y )
5 eluni2 4255 . . 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 3505 1  |-  ( A  =/=  (/)  ->  |^| A  C_  U. A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1819    =/= wne 2652   A.wral 2807   E.wrex 2808    C_ wss 3471   (/)c0 3793   U.cuni 4251   |^|cint 4288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3794  df-uni 4252  df-int 4289
This theorem is referenced by:  unissint  4313  intssuni2  4314  fin23lem31  8740  wunint  9110  tskint  9180  incexc  13661  incexc2  13662  subgint  16352  efgval  16862  lbsextlem3  17933  cssmre  18851  uffixfr  20550  uffix2  20551  uffixsn  20552  insiga  28310  dfon2lem8  29439  intidl  30631  elrfi  30831
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