Theorem wetrep 5107

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 ) )

Proof of Theorem wetrep

Step	Hyp	Ref	Expression
1		weso 5105	. . 3 |- ( _E We A -> _E Or A )
2		sotr 5057	. . 3 |- ( ( _E Or A /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( ( x _E y /\ y _E z ) -> x _E z ) )
3	1, 2	sylan 488	. 2 |- ( ( _E We A /\ ( x e. A /\ y e. A /\ z e. A ) ) -> ( ( x _E y /\ y _E z ) -> x _E z ) )
4		epel 5032	. . 3 |- ( x _E y <-> x e. y )
5		epel 5032	. . 3 |- ( y _E z <-> y e. z )
6	4, 5	anbi12i 733	. 2 |- ( ( x _E y /\ y _E z ) <-> ( x e. y /\ y e. z ) )
7		epel 5032	. 2 |- ( x _E z <-> x e. z )
8	3, 6, 7	3imtr3g 284	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 384   /\ w3a 1037   e. wcel 1990   class class class wbr 4653   _E cep 5028   Or wor 5034   We wwe 5072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-eprel 5029  df-po 5035  df-so 5036  df-we 5075

This theorem is referenced by:  wefrc  5108  ordelord  5745