Theorem chnrun 47909

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 4176	. . . 4 ⊢ 𝑅 ⊆ (𝑅 ∪ < )
2		chnrss 18789	. . . 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 3904	. 2 ⊢ (𝐴 ∈ (𝑅 Chain 𝐵) → 𝐴 ∈ ((𝑅 ∪ < ) Chain 𝐵))
5		ssun2 4177	. . . 4 ⊢ < ⊆ (𝑅 ∪ < )
6		chnrss 18789	. . . 4 ⊢ ( < ⊆ (𝑅 ∪ < ) → ( < Chain 𝐵) ⊆ ((𝑅 ∪ < ) Chain 𝐵))
7	5, 6	ax-mp 5	. . 3 ⊢ ( < Chain 𝐵) ⊆ ((𝑅 ∪ < ) Chain 𝐵)
8	7	sseli 3904	. 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 2144   ⊆ wss 3895   ∪ cun 4149   Chain cchn 18779

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

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  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-ral 3077  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-dm 5660  df-iota 6494  df-fv 6546  df-chn 18780

This theorem is used by:  chnrun2  47910