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Theorem intssunim 3637
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 3309 . . . 4  |-  ( ( E. x  x  e.  A  /\  A. x  e.  A  y  e.  x )  ->  E. x  e.  A  y  e.  x )
21ex 108 . . 3  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  y  e.  x  ->  E. x  e.  A  y  e.  x )
)
3 vex 2560 . . . 4  |-  y  e. 
_V
43elint2 3622 . . 3  |-  ( y  e.  |^| A  <->  A. x  e.  A  y  e.  x )
5 eluni2 3584 . . 3  |-  ( y  e.  U. A  <->  E. x  e.  A  y  e.  x )
62, 4, 53imtr4g 194 . 2  |-  ( E. x  x  e.  A  ->  ( y  e.  |^| A  ->  y  e.  U. A ) )
76ssrdv 2951 1  |-  ( E. x  x  e.  A  ->  |^| A  C_  U. A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1381    e. wcel 1393   A.wral 2306   E.wrex 2307    C_ wss 2917   U.cuni 3580   |^|cint 3615
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-int 3616
This theorem is referenced by:  intssuni2m  3639
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