Theorem ontr2d 36731

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 6413	. . 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 2146   ⊆ wss 3906  Oncon0 6364

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

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 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-tr 5221  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6367  df-on 6368

This theorem is used by:  nadddilem1  36751