Theorem wecmpep 2947

Description: The elements of an epsilon well-ordering are comparable.

Assertion

Ref	Expression
wecmpep	|- ((E We A /\ (x e. A /\ y e. A)) -> (x e. y \/ x = y \/ y e. x))

Proof of Theorem wecmpep

Step	Hyp	Ref	Expression
1		solin 2863	. . 3 |- ((E Or A /\ (x e. A /\ y e. A)) -> (xEy \/ x = y \/ yEx))
2		epel 2840	. . . 4 |- (xEy <-> x e. y)
3		pm4.2 170	. . . 4 |- (x = y <-> x = y)
4		epel 2840	. . . 4 |- (yEx <-> y e. x)
5	2, 3, 4	3orbi123i 825	. . 3 |- ((xEy \/ x = y \/ yEx) <-> (x e. y \/ x = y \/ y e. x))
6	1, 5	sylib 198	. 2 |- ((E Or A /\ (x e. A /\ y e. A)) -> (x e. y \/ x = y \/ y e. x))
7		weso 2946	. 2 |- (E We A -> E Or A)
8	6, 7	sylan 450	1 |- ((E We A /\ (x e. A /\ y e. A)) -> (x e. y \/ x = y \/ y e. x))

Syntax hints:   -> wi 3   /\ wa 223   \/ w3o 776   = wceq 958   e. wcel 960   class class class wbr 2624  Ecep 2836   Or wor 2845   We wwe 2922

This theorem is referenced by:  tz7.7 2979

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-3or 778  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-so 2856  df-we 2940

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