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Theorem unisnif 35906
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 4536 . . . 4 (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = 𝐴)
2 unisng 4929 . . . 4 (𝐴 ∈ V → {𝐴} = 𝐴)
31, 2eqtr4d 2777 . . 3 (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = {𝐴})
4 iffalse 4539 . . . 4 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∅)
5 snprc 4721 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
65biimpi 216 . . . . . 6 𝐴 ∈ V → {𝐴} = ∅)
76unieqd 4924 . . . . 5 𝐴 ∈ V → {𝐴} = ∅)
8 uni0 4939 . . . . 5 ∅ = ∅
97, 8eqtrdi 2790 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
104, 9eqtr4d 2777 . . 3 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = {𝐴})
113, 10pm2.61i 182 . 2 if(𝐴 ∈ V, 𝐴, ∅) = {𝐴}
1211eqcomi 2743 1 {𝐴} = if(𝐴 ∈ V, 𝐴, ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1536  wcel 2105  Vcvv 3477  c0 4338  ifcif 4530  {csn 4630   cuni 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-11 2154  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-uni 4912
This theorem is referenced by:  dfrdg4  35932
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