Theorem unisucs 6441

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 6443. (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 6367	. . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	unieqi 4882	. . 3 ⊢ ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴})
3	2	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = ∪ (𝐴 ∪ {𝐴}))
4		uniun 4893	. . 3 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴})
5	4	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴}))
6		unisng 4888	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
7	6	uneq2d 4118	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴))
8	3, 5, 7	3eqtrd 2801	1 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ∪ cun 3900  {csn 4587  ∪ cuni 4870  suc csuc 6363

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3455  df-un 3907  df-ss 3919  df-sn 4588  df-pr 4590  df-uni 4871  df-suc 6367

This theorem is used by:  unisucg  6442