Theorem wetrep 2932

Description: An epsilon well-ordering is a transitive relation.

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		sotr 2847	. . 3 |- ((E Or A /\ (x e. A /\ y e. A /\ z e. A)) -> ((xEy /\ yEz) -> xEz))
2		weso 2930	. . 3 |- (E We A -> E Or A)
3	1, 2	sylan 448	. 2 |- ((E We A /\ (x e. A /\ y e. A /\ z e. A)) -> ((xEy /\ yEz) -> xEz))
4		epel 2823	. . 3 |- (xEy <-> x e. y)
5		epel 2823	. . 3 |- (yEz <-> y e. z)
6	4, 5	anbi12i 481	. 2 |- ((xEy /\ yEz) <-> (x e. y /\ y e. z))
7		epel 2823	. 2 |- (xEz <-> x e. z)
8	3, 6, 7	3imtr3g 550	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 3   /\ wa 223   /\ w3a 773   e. wcel 955   class class class wbr 2609  Ecep 2819   Or wor 2830   We wwe 2906

This theorem is referenced by:  wefrc 2933  ordelord 2960

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-br 2610  df-opab 2657  df-eprel 2821  df-po 2831  df-so 2841  df-we 2924

Copyright terms: Public domain