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Theorem bj-intabssel1 12578
Description: Version of intss1 3733 using a class abstraction and implicit substitution. Closed form of intmin3 3745. (Contributed by BJ, 29-Nov-2019.)
Hypotheses
Ref Expression
bj-intabssel1.nf  |-  F/_ x A
bj-intabssel1.nf2  |-  F/ x ps
bj-intabssel1.is  |-  ( x  =  A  ->  ( ps  ->  ph ) )
Assertion
Ref Expression
bj-intabssel1  |-  ( A  e.  V  ->  ( ps  ->  |^| { x  | 
ph }  C_  A
) )

Proof of Theorem bj-intabssel1
StepHypRef Expression
1 bj-intabssel1.nf . . 3  |-  F/_ x A
2 bj-intabssel1.nf2 . . 3  |-  F/ x ps
3 bj-intabssel1.is . . 3  |-  ( x  =  A  ->  ( ps  ->  ph ) )
41, 2, 3elabgf2 12568 . 2  |-  ( A  e.  V  ->  ( ps  ->  A  e.  {
x  |  ph }
) )
5 intss1 3733 . 2  |-  ( A  e.  { x  | 
ph }  ->  |^| { x  |  ph }  C_  A
)
64, 5syl6 33 1  |-  ( A  e.  V  ->  ( ps  ->  |^| { x  | 
ph }  C_  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1299   F/wnf 1404    e. wcel 1448   {cab 2086   F/_wnfc 2227    C_ wss 3021   |^|cint 3718
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-in 3027  df-ss 3034  df-int 3719
This theorem is referenced by:  bj-omssind  12718
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