Theorem wrddun2

Description: Superadditivity of word constructor. Class of words over union alphabet includes all words over either alphabet in the union; if B and C are non-empty and different, then this subclass relation is strict because of the words which have symbols both from B and from C . (Contributed by Ender Ting, 24-Jul-2026)

	Ref	Expression
Assertion	wrddun2	⊢ ( Word 𝐵 ∪ Word 𝐶 ) ⊆ Word ( 𝐵 ∪ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		elun	⊢ ( 𝑛 ∈ ( Word 𝐵 ∪ Word 𝐶 ) ↔ ( 𝑛 ∈ Word 𝐵 ∨ 𝑛 ∈ Word 𝐶 ) )
2		wrddun	⊢ ( ( 𝑛 ∈ Word 𝐵 ∨ 𝑛 ∈ Word 𝐶 ) → 𝑛 ∈ Word ( 𝐵 ∪ 𝐶 ) )
3	1 2	sylbi	⊢ ( 𝑛 ∈ ( Word 𝐵 ∪ Word 𝐶 ) → 𝑛 ∈ Word ( 𝐵 ∪ 𝐶 ) )
4	3	ssriv	⊢ ( Word 𝐵 ∪ Word 𝐶 ) ⊆ Word ( 𝐵 ∪ 𝐶 )