Theorem wecmpep 5643

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 5642	. 2 ⊢ ( E We 𝐴 → E Or 𝐴)
2		solin 5586	. . 3 ⊢ (( E Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
3		epel 5554	. . . 4 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
4		biid 264	. . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
5		epel 5554	. . . 4 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
6	3, 4, 5	3orbi123i 1174	. . 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 592	1 ⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ w3o 1102   ∈ wcel 2145   class class class wbr 5103   E cep 5550   Or wor 5558   We wwe 5603

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-eprel 5551  df-so 5560  df-we 5606

This theorem is used by:  tz7.7  6381