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Theorem trintssm 3899
Description: Any inhabited transitive class includes its intersection. Similar to Exercise 3 in [TakeutiZaring] p. 44 (which mistakenly does not include the inhabitedness hypothesis). (Contributed by Jim Kingdon, 22-Aug-2018.)
Assertion
Ref Expression
trintssm  |-  ( ( Tr  A  /\  E. x  x  e.  A
)  ->  |^| A  C_  A )
Distinct variable group:    x, A

Proof of Theorem trintssm
StepHypRef Expression
1 intss1 3659 . . . 4  |-  ( x  e.  A  ->  |^| A  C_  x )
2 trss 3892 . . . . 5  |-  ( Tr  A  ->  ( x  e.  A  ->  x  C_  A ) )
32com12 30 . . . 4  |-  ( x  e.  A  ->  ( Tr  A  ->  x  C_  A ) )
4 sstr2 3007 . . . 4  |-  ( |^| A  C_  x  ->  (
x  C_  A  ->  |^| A  C_  A )
)
51, 3, 4sylsyld 57 . . 3  |-  ( x  e.  A  ->  ( Tr  A  ->  |^| A  C_  A ) )
65exlimiv 1530 . 2  |-  ( E. x  x  e.  A  ->  ( Tr  A  ->  |^| A  C_  A )
)
76impcom 123 1  |-  ( ( Tr  A  /\  E. x  x  e.  A
)  ->  |^| A  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   E.wex 1422    e. wcel 1434    C_ wss 2974   |^|cint 3644   Tr wtr 3883
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-in 2980  df-ss 2987  df-uni 3610  df-int 3645  df-tr 3884
This theorem is referenced by: (None)
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