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Theorem bj-intabssel1 11128
Description: Version of intss1 3686 using a class abstraction and implicit substitution. Closed form of intmin3 3698. (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 11118 . 2  |-  ( A  e.  V  ->  ( ps  ->  A  e.  {
x  |  ph }
) )
5 intss1 3686 . 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 1287   F/wnf 1392    e. wcel 1436   {cab 2071   F/_wnfc 2212    C_ wss 2988   |^|cint 3671
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 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-in 2994  df-ss 3001  df-int 3672
This theorem is referenced by:  bj-omssind  11268
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