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Theorem trintssm 3870
Description: If  A is transitive and inhabited, then  |^| A is a subset of  A. (Contributed by Jim Kingdon, 22-Aug-2018.)
Assertion
Ref Expression
trintssm  |-  ( ( E. x  x  e.  A  /\  Tr  A
)  ->  |^| A  C_  A )
Distinct variable group:    x, A

Proof of Theorem trintssm
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 vex 2560 . . . 4  |-  y  e. 
_V
21elint2 3622 . . 3  |-  ( y  e.  |^| A  <->  A. x  e.  A  y  e.  x )
3 r19.2m 3309 . . . . 5  |-  ( ( E. x  x  e.  A  /\  A. x  e.  A  y  e.  x )  ->  E. x  e.  A  y  e.  x )
43ex 108 . . . 4  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  y  e.  x  ->  E. x  e.  A  y  e.  x )
)
5 trel 3861 . . . . . 6  |-  ( Tr  A  ->  ( (
y  e.  x  /\  x  e.  A )  ->  y  e.  A ) )
65expcomd 1330 . . . . 5  |-  ( Tr  A  ->  ( x  e.  A  ->  ( y  e.  x  ->  y  e.  A ) ) )
76rexlimdv 2432 . . . 4  |-  ( Tr  A  ->  ( E. x  e.  A  y  e.  x  ->  y  e.  A ) )
84, 7sylan9 389 . . 3  |-  ( ( E. x  x  e.  A  /\  Tr  A
)  ->  ( A. x  e.  A  y  e.  x  ->  y  e.  A ) )
92, 8syl5bi 141 . 2  |-  ( ( E. x  x  e.  A  /\  Tr  A
)  ->  ( y  e.  |^| A  ->  y  e.  A ) )
109ssrdv 2951 1  |-  ( ( E. x  x  e.  A  /\  Tr  A
)  ->  |^| A  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97   E.wex 1381    e. wcel 1393   A.wral 2306   E.wrex 2307    C_ wss 2917   |^|cint 3615   Tr wtr 3854
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-ral 2311  df-rex 2312  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-int 3616  df-tr 3855
This theorem is referenced by: (None)
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