Theorem oneltr 43203

Description: The elementhood relation on the ordinals is transitive. Theorem 1.9(ii) of [Schloeder] p. 1. See ontr1 6426. (Contributed by RP, 15-Jan-2025.)

Assertion

Ref	Expression
oneltr	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))

Proof of Theorem oneltr

Step	Hyp	Ref	Expression
1		ontr1 6426	. 2 ⊢ (𝐶 ∈ On → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
2	1	3ad2ant3 1133	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   ∈ wcel 2104  Oncon0 6380

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 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1087  df-tru 1538  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-ral 3058  df-v 3479  df-ss 3980  df-uni 4915  df-tr 5267  df-po 5590  df-so 5591  df-fr 5635  df-we 5637  df-ord 6383  df-on 6384

This theorem is referenced by: (None)