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Theorem intssuni2m 3801
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 3799 . 2  |-  ( E. x  x  e.  A  ->  |^| A  C_  U. A
)
2 uniss 3763 . 2  |-  ( A 
C_  B  ->  U. A  C_ 
U. B )
31, 2sylan9ssr 3114 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 1469    e. wcel 1481    C_ wss 3074   U.cuni 3742   |^|cint 3777
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3080  df-ss 3087  df-uni 3743  df-int 3778
This theorem is referenced by:  rintm  3911  onintonm  4439  fival  6864  fiuni  6872
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