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Theorem trel 4003
Description: In a transitive class, the membership relation is transitive. (Contributed by NM, 19-Apr-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
Assertion
Ref Expression
trel  |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  A )  ->  B  e.  A ) )

Proof of Theorem trel
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 3998 . 2  |-  ( Tr  A  <->  A. y A. x
( ( y  e.  x  /\  x  e.  A )  ->  y  e.  A ) )
2 eleq12 2182 . . . . . 6  |-  ( ( y  =  B  /\  x  =  C )  ->  ( y  e.  x  <->  B  e.  C ) )
3 eleq1 2180 . . . . . . 7  |-  ( x  =  C  ->  (
x  e.  A  <->  C  e.  A ) )
43adantl 275 . . . . . 6  |-  ( ( y  =  B  /\  x  =  C )  ->  ( x  e.  A  <->  C  e.  A ) )
52, 4anbi12d 464 . . . . 5  |-  ( ( y  =  B  /\  x  =  C )  ->  ( ( y  e.  x  /\  x  e.  A )  <->  ( B  e.  C  /\  C  e.  A ) ) )
6 eleq1 2180 . . . . . 6  |-  ( y  =  B  ->  (
y  e.  A  <->  B  e.  A ) )
76adantr 274 . . . . 5  |-  ( ( y  =  B  /\  x  =  C )  ->  ( y  e.  A  <->  B  e.  A ) )
85, 7imbi12d 233 . . . 4  |-  ( ( y  =  B  /\  x  =  C )  ->  ( ( ( y  e.  x  /\  x  e.  A )  ->  y  e.  A )  <->  ( ( B  e.  C  /\  C  e.  A )  ->  B  e.  A ) ) )
98spc2gv 2750 . . 3  |-  ( ( B  e.  C  /\  C  e.  A )  ->  ( A. y A. x ( ( y  e.  x  /\  x  e.  A )  ->  y  e.  A )  ->  (
( B  e.  C  /\  C  e.  A
)  ->  B  e.  A ) ) )
109pm2.43b 52 . 2  |-  ( A. y A. x ( ( y  e.  x  /\  x  e.  A )  ->  y  e.  A )  ->  ( ( B  e.  C  /\  C  e.  A )  ->  B  e.  A ) )
111, 10sylbi 120 1  |-  ( Tr  A  ->  ( ( B  e.  C  /\  C  e.  A )  ->  B  e.  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1314    = wceq 1316    e. wcel 1465   Tr wtr 3996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-in 3047  df-ss 3054  df-uni 3707  df-tr 3997
This theorem is referenced by:  trel3  4004  ordtr1  4280  suctr  4313  trsuc  4314  ordn2lp  4430
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