Theorem unisucs 6396

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 6398. (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 6323	. . . 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 4868	. . 3 ⊢ ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴})
5	4	a1i 11	. 2 ⊢ (𝐴 ∈ 𝑉 → ∪ (𝐴 ∪ {𝐴}) = (∪ 𝐴 ∪ ∪ {𝐴}))
6		unisng 4863	. . 3 ⊢ (𝐴 ∈ 𝑉 → ∪ {𝐴} = 𝐴)
7	6	uneq2d 4105	. 2 ⊢ (𝐴 ∈ 𝑉 → (∪ 𝐴 ∪ ∪ {𝐴}) = (∪ 𝐴 ∪ 𝐴))
8	3, 5, 7	3eqtrd 2779	1 ⊢ (𝐴 ∈ 𝑉 → ∪ suc 𝐴 = (∪ 𝐴 ∪ 𝐴))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119   ∪ cun 3888  {csn 4562  ∪ cuni 4845  suc csuc 6319

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-v 3434  df-un 3895  df-ss 3907  df-sn 4563  df-pr 4565  df-uni 4846  df-suc 6323

This theorem is referenced by:  unisucg  6397