Theorem wecmpep 5644

Description: The elements of a class well-ordered by membership are comparable. (Contributed by NM, 17-May-1994.)

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		weso 5643	. 2 ⊢ ( E We 𝐴 → E Or 𝐴)
2		solin 5587	. . 3 ⊢ (( E Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
3		epel 5555	. . . 4 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
4		biid 264	. . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
5		epel 5555	. . . 4 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
6	3, 4, 5	3orbi123i 1172	. . 3 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
7	2, 6	sylib 221	. 2 ⊢ (( E Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
8	1, 7	sylan 591	1 ⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))

Syntax hints:   → wi 4   ∧ wa 400   ∨ w3o 1100   ∈ wcel 2145   class class class wbr 5105   E cep 5551   Or wor 5559   We wwe 5604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3080  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-eprel 5552  df-so 5561  df-we 5607

This theorem is referenced by:  tz7.7  6376