ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  trintssmOLD Unicode version

Theorem trintssmOLD 3953
Description: Obsolete version of trintssm 3952 as of 30-Oct-2021. (Contributed by Jim Kingdon, 22-Aug-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
trintssmOLD  |-  ( ( E. x  x  e.  A  /\  Tr  A
)  ->  |^| A  C_  A )
Distinct variable group:    x, A

Proof of Theorem trintssmOLD
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 vex 2622 . . . 4  |-  y  e. 
_V
21elint2 3695 . . 3  |-  ( y  e.  |^| A  <->  A. x  e.  A  y  e.  x )
3 r19.2m 3369 . . . . 5  |-  ( ( E. x  x  e.  A  /\  A. x  e.  A  y  e.  x )  ->  E. x  e.  A  y  e.  x )
43ex 113 . . . 4  |-  ( E. x  x  e.  A  ->  ( A. x  e.  A  y  e.  x  ->  E. x  e.  A  y  e.  x )
)
5 trel 3943 . . . . . 6  |-  ( Tr  A  ->  ( (
y  e.  x  /\  x  e.  A )  ->  y  e.  A ) )
65expcomd 1375 . . . . 5  |-  ( Tr  A  ->  ( x  e.  A  ->  ( y  e.  x  ->  y  e.  A ) ) )
76rexlimdv 2488 . . . 4  |-  ( Tr  A  ->  ( E. x  e.  A  y  e.  x  ->  y  e.  A ) )
84, 7sylan9 401 . . 3  |-  ( ( E. x  x  e.  A  /\  Tr  A
)  ->  ( A. x  e.  A  y  e.  x  ->  y  e.  A ) )
92, 8syl5bi 150 . 2  |-  ( ( E. x  x  e.  A  /\  Tr  A
)  ->  ( y  e.  |^| A  ->  y  e.  A ) )
109ssrdv 3031 1  |-  ( ( E. x  x  e.  A  /\  Tr  A
)  ->  |^| A  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   E.wex 1426    e. wcel 1438   A.wral 2359   E.wrex 2360    C_ wss 2999   |^|cint 3688   Tr wtr 3936
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-in 3005  df-ss 3012  df-uni 3654  df-int 3689  df-tr 3937
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator