Theorem wetrep 4406

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

Step	Hyp	Ref	Expression
1		df-3an 982	. . 3 |- ( ( x e. A /\ y e. A /\ z e. A ) <-> ( ( x e. A /\ y e. A ) /\ z e. A ) )
2		df-wetr 4380	. . . . . . . . 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 ) ) )
3	2	simprbi 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 ) )
4	3	r19.21bi 2593	. . . . . . 7 |- ( ( _E We A /\ x e. A ) -> A. y e. A A. z e. A ( ( x _E y /\ y _E z ) -> x _E z ) )
5	4	r19.21bi 2593	. . . . . 6 |- ( ( ( _E We A /\ x e. A ) /\ y e. A ) -> A. z e. A ( ( x _E y /\ y _E z ) -> x _E z ) )
6	5	anasss 399	. . . . 5 |- ( ( _E We A /\ ( x e. A /\ y e. A ) ) -> A. z e. A ( ( x _E y /\ y _E z ) -> x _E z ) )
7	6	r19.21bi 2593	. . . 4 |- ( ( ( _E We A /\ ( x e. A /\ y e. A ) ) /\ z e. A ) -> ( ( x _E y /\ y _E z ) -> x _E z ) )
8	7	anasss 399	. . 3 |- ( ( _E We A /\ ( ( x e. A /\ y e. A ) /\ z e. A ) ) -> ( ( x _E y /\ y _E z ) -> x _E z ) )
9	1, 8	sylan2b 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 4338	. . 3 |- ( x _E y <-> x e. y )
11		epel 4338	. . 3 |- ( y _E z <-> y e. z )
12	10, 11	anbi12i 460	. 2 |- ( ( x _E y /\ y _E z ) <-> ( x e. y /\ y e. z ) )
13		epel 4338	. 2 |- ( x _E z <-> x e. z )
14	9, 12, 13	3imtr3g 204	1 |- ( ( _E We A /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( ( x e. y /\ y e. z ) -> x e. z ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   e. wcel 2175   A.wral 2483   class class class wbr 4043   _E cep 4333   Fr wfr 4374   We wwe 4376

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-br 4044  df-opab 4105  df-eprel 4335  df-wetr 4380

This theorem is referenced by:  wessep  4625