Theorem chnrun 48112

Description: Satisfying either of two chain relations is sufficient to make a chain under their union. (Contributed by Ender Ting, 24-Jul-2026.)

Assertion

Ref	Expression
chnrun	⊢ ((𝐴 ∈ (𝑅 Chain 𝐵) ∨ 𝐴 ∈ ( < Chain 𝐵)) → 𝐴 ∈ ((𝑅 ∪ < ) Chain 𝐵))

Proof of Theorem chnrun

Step	Hyp	Ref	Expression
1		ssun1 4179	. . . 4 ⊢ 𝑅 ⊆ (𝑅 ∪ < )
2		chnrss 18815	. . . 4 ⊢ (𝑅 ⊆ (𝑅 ∪ < ) → (𝑅 Chain 𝐵) ⊆ ((𝑅 ∪ < ) Chain 𝐵))
3	1, 2	ax-mp 5	. . 3 ⊢ (𝑅 Chain 𝐵) ⊆ ((𝑅 ∪ < ) Chain 𝐵)
4	3	sseli 3909	. 2 ⊢ (𝐴 ∈ (𝑅 Chain 𝐵) → 𝐴 ∈ ((𝑅 ∪ < ) Chain 𝐵))
5		ssun2 4180	. . . 4 ⊢ < ⊆ (𝑅 ∪ < )
6		chnrss 18815	. . . 4 ⊢ ( < ⊆ (𝑅 ∪ < ) → ( < Chain 𝐵) ⊆ ((𝑅 ∪ < ) Chain 𝐵))
7	5, 6	ax-mp 5	. . 3 ⊢ ( < Chain 𝐵) ⊆ ((𝑅 ∪ < ) Chain 𝐵)
8	7	sseli 3909	. 2 ⊢ (𝐴 ∈ ( < Chain 𝐵) → 𝐴 ∈ ((𝑅 ∪ < ) Chain 𝐵))
9	4, 8	jaoi 871	1 ⊢ ((𝐴 ∈ (𝑅 Chain 𝐵) ∨ 𝐴 ∈ ( < Chain 𝐵)) → 𝐴 ∈ ((𝑅 ∪ < ) Chain 𝐵))

This proof depends on syntax axioms:   → wi 4   ∨ wo 861   ∈ wcel 2147   ⊆ wss 3900   ∪ cun 4152   Chain cchn 18805

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-or 862  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-ral 3081  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-dm 5665  df-iota 6509  df-fv 6563  df-chn 18806

This theorem is used by:  chnrun2  48113