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Theorem unisnif 34897
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 4535 . . . 4 (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = 𝐴)
2 unisng 4930 . . . 4 (𝐴 ∈ V → {𝐴} = 𝐴)
31, 2eqtr4d 2776 . . 3 (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = {𝐴})
4 iffalse 4538 . . . 4 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∅)
5 snprc 4722 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
65biimpi 215 . . . . . 6 𝐴 ∈ V → {𝐴} = ∅)
76unieqd 4923 . . . . 5 𝐴 ∈ V → {𝐴} = ∅)
8 uni0 4940 . . . . 5 ∅ = ∅
97, 8eqtrdi 2789 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
104, 9eqtr4d 2776 . . 3 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = {𝐴})
113, 10pm2.61i 182 . 2 if(𝐴 ∈ V, 𝐴, ∅) = {𝐴}
1211eqcomi 2742 1 {𝐴} = if(𝐴 ∈ V, 𝐴, ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  Vcvv 3475  c0 4323  ifcif 4529  {csn 4629   cuni 4909
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-uni 4910
This theorem is referenced by:  dfrdg4  34923
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