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Theorem intssuni2m 3853
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 3851 . 2  |-  ( E. x  x  e.  A  ->  |^| A  C_  U. A
)
2 uniss 3815 . 2  |-  ( A 
C_  B  ->  U. A  C_ 
U. B )
31, 2sylan9ssr 3161 1  |-  ( ( A  C_  B  /\  E. x  x  e.  A
)  ->  |^| A  C_  U. B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   E.wex 1485    e. wcel 2141    C_ wss 3121   U.cuni 3794   |^|cint 3829
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-in 3127  df-ss 3134  df-uni 3795  df-int 3830
This theorem is referenced by:  rintm  3963  onintonm  4499  fival  6945  fiuni  6953
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