Theorem oneltri 6358

Description: The elementhood relation on the ordinals is complete, so we have triality. Theorem 1.9(iii) of [Schloeder] p. 1. See ordtri3or 6347. (Contributed by RP, 15-Jan-2025.)

Assertion

Ref	Expression
oneltri	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵))

Proof of Theorem oneltri

Step	Hyp	Ref	Expression
1		eloni 6325	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		eloni 6325	. . 3 ⊢ (𝐵 ∈ On → Ord 𝐵)
3		ordtri3or 6347	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
4	1, 2, 3	syl2an 596	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
5		3orcomb 1093	. 2 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵))
6	4, 5	sylib 218	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∨ w3o 1085   = wceq 1541   ∈ wcel 2113  Ord word 6314  Oncon0 6315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-tr 5204  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-ord 6318  df-on 6319

This theorem is referenced by:  constrfiss  33857  vonf1owev  35251