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Theorem intssunim 3665
Description: The intersection of an inhabited set is a subclass of its union. (Contributed by NM, 29-Jul-2006.)
Assertion
Ref Expression
intssunim  |-  ( E. x  x  e.  A  ->  |^| A  C_  U. A
)
Distinct variable group:    x, A

Proof of Theorem intssunim
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 r19.2m 3337 . . . 4  |-  ( ( E. x  x  e.  A  /\  A. x  e.  A  y  e.  x )  ->  E. x  e.  A  y  e.  x )
21ex 112 . . 3  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  y  e.  x  ->  E. x  e.  A  y  e.  x )
)
3 vex 2577 . . . 4  |-  y  e. 
_V
43elint2 3650 . . 3  |-  ( y  e.  |^| A  <->  A. x  e.  A  y  e.  x )
5 eluni2 3612 . . 3  |-  ( y  e.  U. A  <->  E. x  e.  A  y  e.  x )
62, 4, 53imtr4g 198 . 2  |-  ( E. x  x  e.  A  ->  ( y  e.  |^| A  ->  y  e.  U. A ) )
76ssrdv 2979 1  |-  ( E. x  x  e.  A  ->  |^| A  C_  U. A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1397    e. wcel 1409   A.wral 2323   E.wrex 2324    C_ wss 2945   U.cuni 3608   |^|cint 3643
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-in 2952  df-ss 2959  df-uni 3609  df-int 3644
This theorem is referenced by:  intssuni2m  3667
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