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Theorem wetrep 4357
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 980 . . 3  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  <->  ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A ) )
2 df-wetr 4331 . . . . . . . . 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 275 . . . . . . . 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 2565 . . . . . . 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 2565 . . . . . 6  |-  ( ( (  _E  We  A  /\  x  e.  A
)  /\  y  e.  A )  ->  A. z  e.  A  ( (
x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
65anasss 399 . . . . 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 2565 . . . 4  |-  ( ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A
) )  /\  z  e.  A )  ->  (
( x  _E  y  /\  y  _E  z
)  ->  x  _E  z ) )
87anasss 399 . . 3  |-  ( (  _E  We  A  /\  ( ( x  e.  A  /\  y  e.  A )  /\  z  e.  A ) )  -> 
( ( x  _E  y  /\  y  _E  z )  ->  x  _E  z ) )
91, 8sylan2b 287 . 2  |-  ( (  _E  We  A  /\  ( x  e.  A  /\  y  e.  A  /\  z  e.  A
) )  ->  (
( x  _E  y  /\  y  _E  z
)  ->  x  _E  z ) )
10 epel 4289 . . 3  |-  ( x  _E  y  <->  x  e.  y )
11 epel 4289 . . 3  |-  ( y  _E  z  <->  y  e.  z )
1210, 11anbi12i 460 . 2  |-  ( ( x  _E  y  /\  y  _E  z )  <->  ( x  e.  y  /\  y  e.  z )
)
13 epel 4289 . 2  |-  ( x  _E  z  <->  x  e.  z )
149, 12, 133imtr3g 204 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 104    /\ w3a 978    e. wcel 2148   A.wral 2455   class class class wbr 4000    _E cep 4284    Fr wfr 4325    We wwe 4327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-eprel 4286  df-wetr 4331
This theorem is referenced by:  wessep  4574
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