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Theorem unisnif 32940
 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 4381 . . . 4 (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = 𝐴)
2 unisng 4754 . . . 4 (𝐴 ∈ V → {𝐴} = 𝐴)
31, 2eqtr4d 2832 . . 3 (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = {𝐴})
4 iffalse 4384 . . . 4 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∅)
5 snprc 4554 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
65biimpi 217 . . . . . 6 𝐴 ∈ V → {𝐴} = ∅)
76unieqd 4749 . . . . 5 𝐴 ∈ V → {𝐴} = ∅)
8 uni0 4766 . . . . 5 ∅ = ∅
97, 8syl6eq 2845 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
104, 9eqtr4d 2832 . . 3 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = {𝐴})
113, 10pm2.61i 183 . 2 if(𝐴 ∈ V, 𝐴, ∅) = {𝐴}
1211eqcomi 2802 1 {𝐴} = if(𝐴 ∈ V, 𝐴, ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1520   ∈ wcel 2079  Vcvv 3432  ∅c0 4206  ifcif 4375  {csn 4466  ∪ cuni 4739 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-ext 2767 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-v 3434  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-uni 4740 This theorem is referenced by:  dfrdg4  32966
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