Theorem wetrep 2942

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 2856	. . 3 |- ((E Or A /\ (x e. A /\ y e. A /\ z e. A)) -> ((xEy /\ yEz) -> xEz))
2		weso 2940	. . 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 2834	. . 3 |- (xEy <-> x e. y)
5		epel 2834	. . 3 |- (yEz <-> y e. z)
6	4, 5	anbi12i 482	. 2 |- ((xEy /\ yEz) <-> (x e. y /\ y e. z))
7		epel 2834	. 2 |- (xEz <-> x e. z)
8	3, 6, 7	3imtr3g 552	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 775   e. wcel 958   class class class wbr 2619  Ecep 2830   Or wor 2839   We wwe 2916

This theorem is referenced by:  wefrc 2943  ordelord 2970

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-eprel 2832  df-po 2840  df-so 2850  df-we 2934

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