Theorem dftr5

Description: An alternate way of defining a transitive class. Definition 1.1 of Schloeder p. 1. (Contributed by NM, 20-Mar-2004) Avoid ax-11 . (Revised by BTernaryTau, 28-Dec-2024)

	Ref	Expression
Assertion	dftr5	⊢ ( Tr 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		impexp	⊢ ( ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐴 ) ↔ ( 𝑦 ∈ 𝑥 → ( 𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴 ) ) )
2	1	albii	⊢ ( ∀ 𝑦 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐴 ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ( 𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴 ) ) )
3		df-ral	⊢ ( ∀ 𝑦 ∈ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴 ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ( 𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴 ) ) )
4		r19.21v	⊢ ( ∀ 𝑦 ∈ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴 ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 ) )
5	2 3 4	3bitr2i	⊢ ( ∀ 𝑦 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐴 ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 ) )
6	5	albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐴 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 ) )
7		dftr2c	⊢ ( Tr 𝐴 ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐴 ) )
8		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 ) )
9	6 7 8	3bitr4i	⊢ ( Tr 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 )