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Theorem bj-intabssel1 14627
Description: Version of intss1 3861 using a class abstraction and implicit substitution. Closed form of intmin3 3873. (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 14617 . 2  |-  ( A  e.  V  ->  ( ps  ->  A  e.  {
x  |  ph }
) )
5 intss1 3861 . 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 1353   F/wnf 1460    e. wcel 2148   {cab 2163   F/_wnfc 2306    C_ wss 3131   |^|cint 3846
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-ss 3144  df-int 3847
This theorem is referenced by:  bj-omssind  14772
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