Theorem ontr1i 35786

Description: Transitive law for ordinal numbers. Inference form of ontr1 6425. (Contributed by BTernaryTau, 19-Oct-2026.)

Hypotheses

Ref	Expression
ontr1i.1	⊢ 𝐶 ∈ On
ontr1i.2	⊢ 𝐴 ∈ 𝐵
ontr1i.3	⊢ 𝐵 ∈ 𝐶

Assertion

Ref	Expression
ontr1i	⊢ 𝐴 ∈ 𝐶

Proof of Theorem ontr1i

Step	Hyp	Ref	Expression
1		ontr1i.1	. 2 ⊢ 𝐶 ∈ On
2		ontr1i.2	. . 3 ⊢ 𝐴 ∈ 𝐵
3		ontr1i.3	. . 3 ⊢ 𝐵 ∈ 𝐶
4		ontr1 6425	. . 3 ⊢ (𝐶 ∈ On → ((𝐴 ∈ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
5	2, 3, 4	mp2ani 711	. 2 ⊢ (𝐶 ∈ On → 𝐴 ∈ 𝐶)
6	1, 5	ax-mp 5	1 ⊢ 𝐴 ∈ 𝐶

This proof depends on syntax axioms:   ∈ wcel 2147  Oncon0 6377

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-v 3456  df-ss 3901  df-uni 4871  df-tr 5216  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-ord 6380  df-on 6381

This theorem is used by:  gdlopcfv3  35973  gdlopcfv4  35974  gdlopcfv5  35975  gdlopcfv6  35976  gdlopcfv7  35977  gdlopcfv8  35978