Theorem ontr2d 37164

Description: Transitive law for ordinal numbers. Exercise 3 of [TakeutiZaring] p. 40. Deduction form. (Contributed by Scott Fenton, 31-Jul-2026.)

Hypotheses

Ref	Expression
ontr2d.1	⊢ (𝜑 → 𝐴 ∈ On)
ontr2d.2	⊢ (𝜑 → 𝐶 ∈ On)
ontr2d.3	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
ontr2d.4	⊢ (𝜑 → 𝐵 ∈ 𝐶)

Assertion

Ref	Expression
ontr2d	⊢ (𝜑 → 𝐴 ∈ 𝐶)

Proof of Theorem ontr2d

Step	Hyp	Ref	Expression
1		ontr2d.3	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		ontr2d.4	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝐶)
3		ontr2d.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ On)
4		ontr2d.2	. . 3 ⊢ (𝜑 → 𝐶 ∈ On)
5		ontr2 6437	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
6	3, 4, 5	syl2anc 596	. 2 ⊢ (𝜑 → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
7	1, 2, 6	mp2and 712	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∈ wcel 2147   ⊆ wss 3900  Oncon0 6388

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  ax-sep 5255  ax-pr 5398

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-tr 5219  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6391  df-on 6392

This theorem is used by:  nadddilem1  37184