Theorem unisucs 6357

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 6359. (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 6287	. . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	unieqi 4857	. . 3 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴})
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴}))
4		uniun 4870	. . 3 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴})
5	4	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴}))
6		unisng 4865	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
7	6	uneq2d 4103	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴))
8	3, 5, 7	3eqtrd 2780	1 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104   ∪ cun 3890  {csn 4565  ∪ cuni 4844  suc csuc 6283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3439  df-un 3897  df-in 3899  df-ss 3909  df-sn 4566  df-pr 4568  df-uni 4845  df-suc 6287

This theorem is referenced by:  unisucg  6358