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Theorem unisnif 35926
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 4531 . . . 4 (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = 𝐴)
2 unisng 4925 . . . 4 (𝐴 ∈ V → {𝐴} = 𝐴)
31, 2eqtr4d 2780 . . 3 (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = {𝐴})
4 iffalse 4534 . . . 4 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∅)
5 snprc 4717 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
65biimpi 216 . . . . . 6 𝐴 ∈ V → {𝐴} = ∅)
76unieqd 4920 . . . . 5 𝐴 ∈ V → {𝐴} = ∅)
8 uni0 4935 . . . . 5 ∅ = ∅
97, 8eqtrdi 2793 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
104, 9eqtr4d 2780 . . 3 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = {𝐴})
113, 10pm2.61i 182 . 2 if(𝐴 ∈ V, 𝐴, ∅) = {𝐴}
1211eqcomi 2746 1 {𝐴} = if(𝐴 ∈ V, 𝐴, ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  ifcif 4525  {csn 4626   cuni 4907
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 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-uni 4908
This theorem is referenced by:  dfrdg4  35952
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