Theorem chnrun 47825

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 4124	. . . 4 ⊢ 𝑅 ⊆ (𝑅 ∪ < )
2		chnrss 18736	. . . 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 3927	. 2 ⊢ (𝐴 ∈ (𝑅 Chain 𝐵) → 𝐴 ∈ ((𝑅 ∪ < ) Chain 𝐵))
5		ssun2 4125	. . . 4 ⊢ < ⊆ (𝑅 ∪ < )
6		chnrss 18736	. . . 4 ⊢ ( < ⊆ (𝑅 ∪ < ) → ( < Chain 𝐵) ⊆ ((𝑅 ∪ < ) Chain 𝐵))
7	5, 6	ax-mp 5	. . 3 ⊢ ( < Chain 𝐵) ⊆ ((𝑅 ∪ < ) Chain 𝐵)
8	7	sseli 3927	. 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 2145   ∪ cun 3897   ⊆ wss 3899   Chain cchn 18726

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 2147  ax-9 2155  ax-ext 2732

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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-dm 5658  df-iota 6484  df-fv 6536  df-chn 18727

This theorem is used by:  chnrun2  47826