Theorem unisucs 6364

Description: The union of the successor of a set is equal to the binary union of that set with its union. (Contributed by NM, 30-Aug-1993.) Extract from unisuc 6366. (Revised by BJ, 28-Dec-2024.)

Assertion

Ref	Expression
unisucs	⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴))

Proof of Theorem unisucs

Step	Hyp	Ref	Expression
1		df-suc 6294	. . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	unieqi 4863	. . 3 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴})
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴}))
4		uniun 4876	. . 3 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴})
5	4	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴}))
6		unisng 4871	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
7	6	uneq2d 4108	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴))
8	3, 5, 7	3eqtrd 2781	1 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105   ∪ cun 3895  {csn 4571  ∪ cuni 4850  suc csuc 6290

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 2708

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3443  df-un 3902  df-in 3904  df-ss 3914  df-sn 4572  df-pr 4574  df-uni 4851  df-suc 6294

This theorem is referenced by:  unisucg  6365