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Theorem unisnif 33802
 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
StepHypRef Expression
1 iftrue 4429 . . . 4 (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = 𝐴)
2 unisng 4822 . . . 4 (𝐴 ∈ V → {𝐴} = 𝐴)
31, 2eqtr4d 2796 . . 3 (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = {𝐴})
4 iffalse 4432 . . . 4 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∅)
5 snprc 4613 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
65biimpi 219 . . . . . 6 𝐴 ∈ V → {𝐴} = ∅)
76unieqd 4815 . . . . 5 𝐴 ∈ V → {𝐴} = ∅)
8 uni0 4831 . . . . 5 ∅ = ∅
97, 8eqtrdi 2809 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
104, 9eqtr4d 2796 . . 3 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = {𝐴})
113, 10pm2.61i 185 . 2 if(𝐴 ∈ V, 𝐴, ∅) = {𝐴}
1211eqcomi 2767 1 {𝐴} = if(𝐴 ∈ V, 𝐴, ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1538   ∈ wcel 2111  Vcvv 3409  ∅c0 4227  ifcif 4423  {csn 4525  ∪ cuni 4801 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-11 2158  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-ral 3075  df-rex 3076  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-uni 4802 This theorem is referenced by:  dfrdg4  33828
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