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Theorem intminss 3640
Description: Under subset ordering, the intersection of a restricted class abstraction is less than or equal to any of its members. (Contributed by NM, 7-Sep-2013.)
Hypothesis
Ref Expression
intminss.1  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
Assertion
Ref Expression
intminss  |-  ( ( A  e.  B  /\  ps )  ->  |^| { x  e.  B  |  ph }  C_  A )
Distinct variable groups:    x, A    x, B    ps, x
Allowed substitution hint:    ph( x)

Proof of Theorem intminss
StepHypRef Expression
1 intminss.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ps ) )
21elrab 2698 . 2  |-  ( A  e.  { x  e.  B  |  ph }  <->  ( A  e.  B  /\  ps ) )
3 intss1 3630 . 2  |-  ( A  e.  { x  e.  B  |  ph }  ->  |^| { x  e.  B  |  ph }  C_  A )
42, 3sylbir 125 1  |-  ( ( A  e.  B  /\  ps )  ->  |^| { x  e.  B  |  ph }  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    <-> wb 98    = wceq 1243    e. wcel 1393   {crab 2310    C_ wss 2917   |^|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-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-int 3616
This theorem is referenced by:  onintss  4127  cardonle  6367
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