Theorem unisuc 6342

Description: A transitive class is equal to the union of its successor. Combines Theorem 4E of [Enderton] p. 72 and Exercise 6 of [Enderton] p. 73. (Contributed by NM, 30-Aug-1993.)

Hypothesis

Ref	Expression
unisuc.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
unisuc	⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)

Proof of Theorem unisuc

Step	Hyp	Ref	Expression
1		ssequn1 4114	. 2 ⊢ (∪ 𝐴 ⊆ 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
2		df-tr 5192	. 2 ⊢ (Tr 𝐴 ↔ ∪ 𝐴 ⊆ 𝐴)
3		df-suc 6272	. . . . 5 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
4	3	unieqi 4852	. . . 4 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴})
5		uniun 4864	. . . 4 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴})
6		unisuc.1	. . . . . 6 ⊢ 𝐴 ∈ V
7	6	unisn 4861	. . . . 5 ⊢ ∪ {𝐴} = 𝐴
8	7	uneq2i 4094	. . . 4 ⊢ (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴)
9	4, 5, 8	3eqtri 2770	. . 3 ⊢ ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴)
10	9	eqeq1i 2743	. 2 ⊢ (∪ suc 𝐴 = 𝐴 ↔ (∪ 𝐴 ∪ 𝐴) = 𝐴)
11	1, 2, 10	3bitr4i 303	1 ⊢ (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴)

Syntax hints:   ↔ wb 205   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887  {csn 4561  ∪ cuni 4839  Tr wtr 5191  suc csuc 6268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892  df-in 3894  df-ss 3904  df-sn 4562  df-pr 4564  df-uni 4840  df-tr 5192  df-suc 6272

This theorem is referenced by:  onunisuci  6380  ordunisuc  7679