Theorem dftr5 5269

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 2153. (Revised by BTernaryTau, 28-Dec-2024.)

Assertion

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

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem dftr5

Step	Hyp	Ref	Expression
1		impexp 450	. . . . 5 ⊢ (((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝑥 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴)))
2	1	albii 1820	. . . 4 ⊢ (∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) ↔ ∀𝑦(𝑦 ∈ 𝑥 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴)))
3		df-ral 3061	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴) ↔ ∀𝑦(𝑦 ∈ 𝑥 → (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴)))
4		r19.21v 3178	. . . 4 ⊢ (∀𝑦 ∈ 𝑥 (𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴))
5	2, 3, 4	3bitr2i 299	. . 3 ⊢ (∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴))
6	5	albii 1820	. 2 ⊢ (∀𝑥∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴) ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴))
7		dftr2c 5268	. 2 ⊢ (Tr 𝐴 ↔ ∀𝑥∀𝑦((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
8		df-ral 3061	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴))
9	6, 7, 8	3bitr4i 303	1 ⊢ (Tr 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1538   ∈ wcel 2105  ∀wral 3060  Tr wtr 5265

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-v 3475  df-in 3955  df-ss 3965  df-uni 4909  df-tr 5266

This theorem is referenced by:  dftr3  5271  smobeth  10587  oaun3lem1  42587