Theorem unisucs 6442

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 6444. (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 6371	. . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	unieqi 4922	. . 3 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴})
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴}))
4		uniun 4935	. . 3 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴})
5	4	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴}))
6		unisng 4930	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
7	6	uneq2d 4164	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴))
8	3, 5, 7	3eqtrd 2777	1 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107   ∪ cun 3947  {csn 4629  ∪ cuni 4909  suc csuc 6367

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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-in 3956  df-ss 3966  df-sn 4630  df-pr 4632  df-uni 4910  df-suc 6371

This theorem is referenced by:  unisucg  6443