Theorem chnrun2 47713

Description: Superadditivity of chain constructor over relation parameter. (Contributed by Ender Ting, 24-Jul-2026.)

Assertion

Ref	Expression
chnrun2	⊢ ((𝑅 Chain 𝐵) ∪ ( < Chain 𝐵)) ⊆ ((𝑅 ∪ < ) Chain 𝐵)

Proof of Theorem chnrun2

Dummy variable 𝑛 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elun 4103	. . 3 ⊢ (𝑛 ∈ ((𝑅 Chain 𝐵) ∪ ( < Chain 𝐵)) ↔ (𝑛 ∈ (𝑅 Chain 𝐵) ∨ 𝑛 ∈ ( < Chain 𝐵)))
2		chnrun 47712	. . 3 ⊢ ((𝑛 ∈ (𝑅 Chain 𝐵) ∨ 𝑛 ∈ ( < Chain 𝐵)) → 𝑛 ∈ ((𝑅 ∪ < ) Chain 𝐵))
3	1, 2	sylbi 220	. 2 ⊢ (𝑛 ∈ ((𝑅 Chain 𝐵) ∪ ( < Chain 𝐵)) → 𝑛 ∈ ((𝑅 ∪ < ) Chain 𝐵))
4	3	ssriv 3938	1 ⊢ ((𝑅 Chain 𝐵) ∪ ( < Chain 𝐵)) ⊆ ((𝑅 ∪ < ) Chain 𝐵)

This proof depends on syntax axioms:   ∨ wo 861   ∈ wcel 2145   ∪ cun 3900   ⊆ wss 3902   Chain cchn 18695

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 2734

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 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-dm 5669  df-iota 6493  df-fv 6545  df-chn 18696

This theorem is used by: (None)