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Theorem intssuni2m 3712
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 3710 . 2  |-  ( E. x  x  e.  A  ->  |^| A  C_  U. A
)
2 uniss 3674 . 2  |-  ( A 
C_  B  ->  U. A  C_ 
U. B )
31, 2sylan9ssr 3039 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 1426    e. wcel 1438    C_ wss 2999   U.cuni 3653   |^|cint 3688
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654  df-int 3689
This theorem is referenced by:  rintm  3821  onintonm  4334
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