Theorem ontr2d 36692

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 6409	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐶 ∈ On) → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
6	3, 4, 5	syl2anc 595	. 2 ⊢ (𝜑 → ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐶) → 𝐴 ∈ 𝐶))
7	1, 2, 6	mp2and 711	1 ⊢ (𝜑 → 𝐴 ∈ 𝐶)

Syntax hints:   → wi 4   ∧ wa 400   ∈ wcel 2143   ⊆ wss 3905  Oncon0 6360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-opab 5174  df-tr 5219  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-ord 6363  df-on 6364

This theorem is referenced by:  nadddilem1  36712