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Theorem intssuni 4072
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 3717 . . . 4  |-  ( ( A  =/=  (/)  /\  A. y  e.  A  x  e.  y )  ->  E. y  e.  A  x  e.  y )
21ex 424 . . 3  |-  ( A  =/=  (/)  ->  ( A. y  e.  A  x  e.  y  ->  E. y  e.  A  x  e.  y ) )
3 vex 2959 . . . 4  |-  x  e. 
_V
43elint2 4057 . . 3  |-  ( x  e.  |^| A  <->  A. y  e.  A  x  e.  y )
5 eluni2 4019 . . 3  |-  ( x  e.  U. A  <->  E. y  e.  A  x  e.  y )
62, 4, 53imtr4g 262 . 2  |-  ( A  =/=  (/)  ->  ( x  e.  |^| A  ->  x  e.  U. A ) )
76ssrdv 3354 1  |-  ( A  =/=  (/)  ->  |^| A  C_  U. A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    =/= wne 2599   A.wral 2705   E.wrex 2706    C_ wss 3320   (/)c0 3628   U.cuni 4015   |^|cint 4050
This theorem is referenced by:  unissint  4074  intssuni2  4075  fin23lem31  8223  wunint  8590  tskint  8660  incexc  12617  incexc2  12618  subgint  14964  efgval  15349  lbsextlem3  16232  cssmre  16920  uffixfr  17955  uffix2  17956  uffixsn  17957  insiga  24520  dfon2lem8  25417  intidl  26639  elrfi  26748
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629  df-uni 4016  df-int 4051
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