Theorem dftr5

Description: An alternate way of defining a transitive class. (Contributed by NM, 20-Mar-2004)

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

Proof

Step	Hyp	Ref	Expression
1		dftr2	⊢ ( Tr 𝐴 ↔ ∀ 𝑦 ∀ 𝑥 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐴 ) )
2		alcom	⊢ ( ∀ 𝑦 ∀ 𝑥 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐴 ) ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐴 ) )
3		impexp	⊢ ( ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐴 ) ↔ ( 𝑦 ∈ 𝑥 → ( 𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴 ) ) )
4	3	albii	⊢ ( ∀ 𝑦 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐴 ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ( 𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴 ) ) )
5		df-ral	⊢ ( ∀ 𝑦 ∈ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴 ) ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → ( 𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴 ) ) )
6	4 5	bitr4i	⊢ ( ∀ 𝑦 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐴 ) ↔ ∀ 𝑦 ∈ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴 ) )
7		r19.21v	⊢ ( ∀ 𝑦 ∈ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴 ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 ) )
8	6 7	bitri	⊢ ( ∀ 𝑦 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐴 ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 ) )
9	8	albii	⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐴 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 ) )
10		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 ) )
11	9 10	bitr4i	⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐴 ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 )
12	2 11	bitri	⊢ ( ∀ 𝑦 ∀ 𝑥 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐴 ) ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 )
13	1 12	bitri	⊢ ( Tr 𝐴 ↔ ∀ 𝑥 ∈ 𝐴 ∀ 𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 )