Theorem oneltri 6406

Description: The elementhood relation on the ordinals is complete, so we have triality. Theorem 1.9(iii) of [Schloeder] p. 1. See ordtri3or 6395. (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 6372	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		eloni 6372	. . 3 ⊢ (𝐵 ∈ On → Ord 𝐵)
3		ordtri3or 6395	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
4	1, 2, 3	syl2an 608	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
5		3orcomb 1110	. 2 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵))
6	4, 5	sylib 221	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ w3o 1102   = wceq 1570   ∈ wcel 2145  Ord word 6361  Oncon0 6362

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-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6365  df-on 6366

This theorem is used by:  constrfiss  34383  vonf1wev  35887  vonf1owevOLD  35889