Theorem wecmpep 5302

Description: The elements of an epsilon well-ordering 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 5301	. 2 ⊢ ( E We 𝐴 → E Or 𝐴)
2		solin 5254	. . 3 ⊢ (( E Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
3		epel 5226	. . . 4 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
4		biid 253	. . . 4 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
5		epel 5226	. . . 4 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
6	3, 4, 5	3orbi123i 1196	. . 3 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
7	2, 6	sylib 210	. 2 ⊢ (( E Or 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
8	1, 7	sylan 576	1 ⊢ (( E We 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))

Syntax hints:   → wi 4   ∧ wa 385   ∨ w3o 1107   ∈ wcel 2157   class class class wbr 4841   E cep 5222   Or wor 5230   We wwe 5268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rab 3096  df-v 3385  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-sn 4367  df-pr 4369  df-op 4373  df-br 4842  df-opab 4904  df-eprel 5223  df-so 5232  df-we 5271

This theorem is referenced by:  tz7.7  5965