Theorem wetrep 2948

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 2862	. . 3 |- ((E Or A /\ (x e. A /\ y e. A /\ z e. A)) -> ((xEy /\ yEz) -> xEz))
2		weso 2946	. . 3 |- (E We A -> E Or A)
3	1, 2	sylan 450	. 2 |- ((E We A /\ (x e. A /\ y e. A /\ z e. A)) -> ((xEy /\ yEz) -> xEz))
4		epel 2840	. . 3 |- (xEy <-> x e. y)
5		epel 2840	. . 3 |- (yEz <-> y e. z)
6	4, 5	anbi12i 484	. 2 |- ((xEy /\ yEz) <-> (x e. y /\ y e. z))
7		epel 2840	. 2 |- (xEz <-> x e. z)
8	3, 6, 7	3imtr3g 554	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 777   e. wcel 960   class class class wbr 2624  Ecep 2836   Or wor 2845   We wwe 2922

This theorem is referenced by:  wefrc 2949  ordelord 2976

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-eprel 2838  df-po 2846  df-so 2856  df-we 2940

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