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Theorem intssuni2m 3681
Description: Subclass relationship for intersection and union. (Contributed by Jim Kingdon, 14-Aug-2018.)
Assertion
Ref Expression
intssuni2m  |-  ( ( A  C_  B  /\  E. x  x  e.  A
)  ->  |^| A  C_  U. B )
Distinct variable group:    x, A
Allowed substitution hint:    B( x)

Proof of Theorem intssuni2m
StepHypRef Expression
1 intssunim 3679 . 2  |-  ( E. x  x  e.  A  ->  |^| A  C_  U. A
)
2 uniss 3643 . 2  |-  ( A 
C_  B  ->  U. A  C_ 
U. B )
31, 2sylan9ssr 3023 1  |-  ( ( A  C_  B  /\  E. x  x  e.  A
)  ->  |^| A  C_  U. B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   E.wex 1422    e. wcel 1434    C_ wss 2983   U.cuni 3622   |^|cint 3657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2613  df-in 2989  df-ss 2996  df-uni 3623  df-int 3658
This theorem is referenced by:  rintm  3786  onintonm  4290
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