Theorem unisucg 6397

Description: A transitive class is equal to the union of its successor, closed form. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.) Generalize from unisuc 6398. (Revised by BJ, 28-Dec-2024.)

Assertion

Ref	Expression
unisucg	⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))

Proof of Theorem unisucg

Step	Hyp	Ref	Expression
1		ssequn1 4138	. . 3 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
2	1	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ⊆ 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴))
3		df-tr 5206	. . 3 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
4	3	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴))
5		unisucs 6396	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴))
6	5	eqeq1d 2738	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ suc 𝐴 = 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴))
7	2, 4, 6	3bitr4d 311	1 ⊢ (𝐴 ∈ 𝑉 → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113   ∪ cun 3899   ⊆ wss 3901  ∪ cuni 4863  Tr wtr 5205  suc csuc 6319

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-un 3906  df-ss 3918  df-sn 4581  df-pr 4583  df-uni 4864  df-tr 5206  df-suc 6323

This theorem is referenced by:  unisuc  6398  onunisuc  6429