Theorem ontr2d 37073

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 6414	. . 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 2144   ⊆ wss 3896  Oncon0 6365

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-pss 4013  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6368  df-on 6369

This theorem is used by:  nadddilem1  37093