Theorem unisnif 35889

Description: Express union of singleton in terms of if. (Contributed by Scott Fenton, 27-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
unisnif	⊢ ∪ {𝐴} = if(𝐴 ∈ V, 𝐴, ∅)

Proof of Theorem unisnif

Step	Hyp	Ref	Expression
1		iftrue 4506	. . . 4 ⊢ (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = 𝐴)
2		unisng 4901	. . . 4 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴)
3	1, 2	eqtr4d 2773	. . 3 ⊢ (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴})
4		iffalse 4509	. . . 4 ⊢ (¬ 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∅)
5		snprc 4693	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
6	5	biimpi 216	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
7	6	unieqd 4896	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∪ ∅)
8		uni0 4911	. . . . 5 ⊢ ∪ ∅ = ∅
9	7, 8	eqtrdi 2786	. . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∅)
10	4, 9	eqtr4d 2773	. . 3 ⊢ (¬ 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴})
11	3, 10	pm2.61i 182	. 2 ⊢ if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴}
12	11	eqcomi 2744	1 ⊢ ∪ {𝐴} = if(𝐴 ∈ V, 𝐴, ∅)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2108  Vcvv 3459  ∅c0 4308  ifcif 4500  {csn 4601  ∪ cuni 4883

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-uni 4884

This theorem is referenced by:  dfrdg4  35915