Theorem wecmpep 2947

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

Assertion

Ref	Expression
wecmpep	⊢ ((E We A ⋀ (x ∈ A ⋀ y ∈ A)) → (x ∈ y ⋁ x = y ⋁ y ∈ x))

Proof of Theorem wecmpep

Step	Hyp	Ref	Expression
1		solin 2863	. . 3 ⊢ ((E Or A ⋀ (x ∈ A ⋀ y ∈ A)) → (xEy ⋁ x = y ⋁ yEx))
2		epel 2840	. . . 4 ⊢ (xEy ↔ x ∈ y)
3		pm4.2 170	. . . 4 ⊢ (x = y ↔ x = y)
4		epel 2840	. . . 4 ⊢ (yEx ↔ y ∈ x)
5	2, 3, 4	3orbi123i 825	. . 3 ⊢ ((xEy ⋁ x = y ⋁ yEx) ↔ (x ∈ y ⋁ x = y ⋁ y ∈ x))
6	1, 5	sylib 198	. 2 ⊢ ((E Or A ⋀ (x ∈ A ⋀ y ∈ A)) → (x ∈ y ⋁ x = y ⋁ y ∈ x))
7		weso 2946	. 2 ⊢ (E We A → E Or A)
8	6, 7	sylan 450	1 ⊢ ((E We A ⋀ (x ∈ A ⋀ y ∈ A)) → (x ∈ y ⋁ x = y ⋁ y ∈ x))

Syntax hints:   → wi 3   ⋀ wa 223   ⋁ w3o 776   = wceq 958   ∈ 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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