Theorem unisnif 36136

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 4487	. . . 4 ⊢ (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = 𝐴)
2		unisng 4883	. . . 4 ⊢ (𝐴 ∈ V → ∪ {𝐴} = 𝐴)
3	1, 2	eqtr4d 2775	. . 3 ⊢ (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴})
4		iffalse 4490	. . . 4 ⊢ (¬ 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∅)
5		snprc 4676	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
6	5	biimpi 216	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → {𝐴} = ∅)
7	6	unieqd 4878	. . . . 5 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∪ ∅)
8		uni0 4893	. . . . 5 ⊢ ∪ ∅ = ∅
9	7, 8	eqtrdi 2788	. . . 4 ⊢ (¬ 𝐴 ∈ V → ∪ {𝐴} = ∅)
10	4, 9	eqtr4d 2775	. . 3 ⊢ (¬ 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴})
11	3, 10	pm2.61i 182	. 2 ⊢ if(𝐴 ∈ V, 𝐴, ∅) = ∪ {𝐴}
12	11	eqcomi 2746	1 ⊢ ∪ {𝐴} = if(𝐴 ∈ V, 𝐴, ∅)

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2114  Vcvv 3442  ∅c0 4287  ifcif 4481  {csn 4582  ∪ cuni 4865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-uni 4866

This theorem is referenced by:  dfrdg4  36164