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Theorem wetrep 4143
Description: An epsilon well-ordering is a transitive relation. (Contributed by NM, 22-Apr-1994.)
Assertion
Ref Expression
wetrep  |-  ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A
) )  ->  (
( x  e.  y  /\  y  e.  z )  ->  x  e.  z ) )
Distinct variable group:    x, A, y, z

Proof of Theorem wetrep
StepHypRef Expression
1 df-3an 922 . . 3  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  <->  ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A ) )
2 df-wetr 4117 . . . . . . . . 9  |-  (  _E  We  A  <->  (  _E  Fr  A  /\  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( (
x  _E  y  /\  y  _E  z )  ->  x  _E  z ) ) )
32simprbi 269 . . . . . . . 8  |-  (  _E  We  A  ->  A. x  e.  A  A. y  e.  A  A. z  e.  A  ( (
x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
43r19.21bi 2454 . . . . . . 7  |-  ( (  _E  We  A  /\  x  e.  A )  ->  A. y  e.  A  A. z  e.  A  ( ( x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
54r19.21bi 2454 . . . . . 6  |-  ( ( (  _E  We  A  /\  x  e.  A
)  /\  y  e.  A )  ->  A. z  e.  A  ( (
x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
65anasss 391 . . . . 5  |-  ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A
) )  ->  A. z  e.  A  ( (
x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
76r19.21bi 2454 . . . 4  |-  ( ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  (
( x  _E  y  /\  y  _E  z
)  ->  x  _E  z ) )
87anasss 391 . . 3  |-  ( (  _E  We  A  /\  ( ( x  e.  A  /\  y  e.  A )  /\  z  e.  A ) )  -> 
( ( x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
91, 8sylan2b 281 . 2  |-  ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A
) )  ->  (
( x  _E  y  /\  y  _E  z
)  ->  x  _E  z ) )
10 epel 4075 . . 3  |-  ( x  _E  y  <->  x  e.  y )
11 epel 4075 . . 3  |-  ( y  _E  z  <->  y  e.  z )
1210, 11anbi12i 448 . 2  |-  ( ( x  _E  y  /\  y  _E  z )  <->  ( x  e.  y  /\  y  e.  z )
)
13 epel 4075 . 2  |-  ( x  _E  z  <->  x  e.  z )
149, 12, 133imtr3g 202 1  |-  ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A
) )  ->  (
( x  e.  y  /\  y  e.  z )  ->  x  e.  z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    e. wcel 1434   A.wral 2353   class class class wbr 3805    _E cep 4070    Fr wfr 4111    We wwe 4113
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-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-eprel 4072  df-wetr 4117
This theorem is referenced by:  wessep  4348
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