Theorem oneltri 42957

Description: The elementhood relation on the ordinals is complete, so we have triality. Theorem 1.9(iii) of [Schloeder] p. 1. See ordtri3or 6397. (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 6375	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		eloni 6375	. . 3 ⊢ (𝐵 ∈ On → Ord 𝐵)
3		ordtri3or 6397	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
4	1, 2, 3	syl2an 594	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))
5		3orcomb 1091	. 2 ⊢ ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵))
6	4, 5	sylib 217	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∈ 𝐵 ∨ 𝐵 ∈ 𝐴 ∨ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∧ wa 394   ∨ w3o 1083   = wceq 1534   ∈ wcel 2099  Ord word 6364  Oncon0 6365

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5144  df-opab 5206  df-tr 5261  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6368  df-on 6369

This theorem is referenced by: (None)