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Theorem wetrep 4178
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 926 . . 3  |-  ( ( x  e.  A  /\  y  e.  A  /\  z  e.  A )  <->  ( ( x  e.  A  /\  y  e.  A
)  /\  z  e.  A ) )
2 df-wetr 4152 . . . . . . . . 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 2461 . . . . . . 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 2461 . . . . . 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 2461 . . . 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 4110 . . 3  |-  ( x  _E  y  <->  x  e.  y )
11 epel 4110 . . 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 4110 . 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 924    e. wcel 1438   A.wral 2359   class class class wbr 3837    _E cep 4105    Fr wfr 4146    We wwe 4148
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-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-eprel 4107  df-wetr 4152
This theorem is referenced by:  wessep  4383
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