Theorem wecmpep 2936

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 2852	. . 3 |- ((E Or A /\ (x e. A /\ y e. A)) -> (xEy \/ x = y \/ yEx))
2		epel 2829	. . . 4 |- (xEy <-> x e. y)
3		pm4.2 170	. . . 4 |- (x = y <-> x = y)
4		epel 2829	. . . 4 |- (yEx <-> y e. x)
5	2, 3, 4	3orbi123i 822	. . 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 2935	. 2 |- (E We A -> E Or A)
8	6, 7	sylan 448	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 773   = wceq 954   e. wcel 956   class class class wbr 2614  Ecep 2825   Or wor 2834   We wwe 2911

This theorem is referenced by:  tz7.7 2968

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2615  df-opab 2662  df-eprel 2827  df-so 2845  df-we 2929

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