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Theorem unisnif 31727
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 4070 . . . 4 (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = 𝐴)
2 unisng 4425 . . . 4 (𝐴 ∈ V → {𝐴} = 𝐴)
31, 2eqtr4d 2658 . . 3 (𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = {𝐴})
4 iffalse 4073 . . . 4 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = ∅)
5 snprc 4230 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
65biimpi 206 . . . . . 6 𝐴 ∈ V → {𝐴} = ∅)
76unieqd 4419 . . . . 5 𝐴 ∈ V → {𝐴} = ∅)
8 uni0 4438 . . . . 5 ∅ = ∅
97, 8syl6eq 2671 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
104, 9eqtr4d 2658 . . 3 𝐴 ∈ V → if(𝐴 ∈ V, 𝐴, ∅) = {𝐴})
113, 10pm2.61i 176 . 2 if(𝐴 ∈ V, 𝐴, ∅) = {𝐴}
1211eqcomi 2630 1 {𝐴} = if(𝐴 ∈ V, 𝐴, ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  Vcvv 3190  c0 3897  ifcif 4064  {csn 4155   cuni 4409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-uni 4410
This theorem is referenced by:  dfrdg4  31753
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