Theorem chninf

Description: There is an infinite number of chains for any infinite alphabet and any relation. For instance, all the singletons of alphabet characters match. (Contributed by Ender Ting, 20-Jan-2026)

	Ref	Expression
Assertion	chninf	⊢ ( 𝐴 ∉ Fin → ( < Chain 𝐴 ) ∉ Fin )

Proof

Step	Hyp	Ref	Expression
1		id	⊢ ( 𝑦 ∈ 𝐴 → 𝑦 ∈ 𝐴 )
2	1	s1chn	⊢ ( 𝑦 ∈ 𝐴 → ⟨“ 𝑦 ”⟩ ∈ ( < Chain 𝐴 ) )
3	2	rgen	⊢ ∀ 𝑦 ∈ 𝐴 ⟨“ 𝑦 ”⟩ ∈ ( < Chain 𝐴 )
4		s111	⊢ ( ( 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( ⟨“ 𝑦 ”⟩ = ⟨“ 𝑥 ”⟩ ↔ 𝑦 = 𝑥 ) )
5	4	biimpd	⊢ ( ( 𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( ⟨“ 𝑦 ”⟩ = ⟨“ 𝑥 ”⟩ → 𝑦 = 𝑥 ) )
6	5	rgen2	⊢ ∀ 𝑦 ∈ 𝐴 ∀ 𝑥 ∈ 𝐴 ( ⟨“ 𝑦 ”⟩ = ⟨“ 𝑥 ”⟩ → 𝑦 = 𝑥 )
7	3 6	pm3.2i	⊢ ( ∀ 𝑦 ∈ 𝐴 ⟨“ 𝑦 ”⟩ ∈ ( < Chain 𝐴 ) ∧ ∀ 𝑦 ∈ 𝐴 ∀ 𝑥 ∈ 𝐴 ( ⟨“ 𝑦 ”⟩ = ⟨“ 𝑥 ”⟩ → 𝑦 = 𝑥 ) )
8		eqid	⊢ ( 𝑦 ∈ 𝐴 ↦ ⟨“ 𝑦 ”⟩ ) = ( 𝑦 ∈ 𝐴 ↦ ⟨“ 𝑦 ”⟩ )
9		s1eq	⊢ ( 𝑦 = 𝑥 → ⟨“ 𝑦 ”⟩ = ⟨“ 𝑥 ”⟩ )
10	8 9	f1mpt	⊢ ( ( 𝑦 ∈ 𝐴 ↦ ⟨“ 𝑦 ”⟩ ) : 𝐴 –1-1→ ( < Chain 𝐴 ) ↔ ( ∀ 𝑦 ∈ 𝐴 ⟨“ 𝑦 ”⟩ ∈ ( < Chain 𝐴 ) ∧ ∀ 𝑦 ∈ 𝐴 ∀ 𝑥 ∈ 𝐴 ( ⟨“ 𝑦 ”⟩ = ⟨“ 𝑥 ”⟩ → 𝑦 = 𝑥 ) ) )
11	7 10	mpbir	⊢ ( 𝑦 ∈ 𝐴 ↦ ⟨“ 𝑦 ”⟩ ) : 𝐴 –1-1→ ( < Chain 𝐴 )
12		f1fi	⊢ ( ( ( < Chain 𝐴 ) ∈ Fin ∧ ( 𝑦 ∈ 𝐴 ↦ ⟨“ 𝑦 ”⟩ ) : 𝐴 –1-1→ ( < Chain 𝐴 ) ) → 𝐴 ∈ Fin )
13	11 12	mpan2	⊢ ( ( < Chain 𝐴 ) ∈ Fin → 𝐴 ∈ Fin )
14	13	a1i	⊢ ( ⊤ → ( ( < Chain 𝐴 ) ∈ Fin → 𝐴 ∈ Fin ) )
15	14	nelcon3d	⊢ ( ⊤ → ( 𝐴 ∉ Fin → ( < Chain 𝐴 ) ∉ Fin ) )
16	15	mptru	⊢ ( 𝐴 ∉ Fin → ( < Chain 𝐴 ) ∉ Fin )