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Theorem unisn2 5303
Description: A version of unisn 4921 without the 𝐴 ∈ V hypothesis. (Contributed by Stefan Allan, 14-Mar-2006.)
Assertion
Ref Expression
unisn2 {𝐴} ∈ {∅, 𝐴}

Proof of Theorem unisn2
StepHypRef Expression
1 unisng 4920 . . 3 (𝐴 ∈ V → {𝐴} = 𝐴)
2 prid2g 4758 . . 3 (𝐴 ∈ V → 𝐴 ∈ {∅, 𝐴})
31, 2eqeltrd 2825 . 2 (𝐴 ∈ V → {𝐴} ∈ {∅, 𝐴})
4 snprc 4714 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
54biimpi 215 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
65unieqd 4913 . . 3 𝐴 ∈ V → {𝐴} = ∅)
7 uni0 4930 . . . 4 ∅ = ∅
8 0ex 5298 . . . . 5 ∅ ∈ V
98prid1 4759 . . . 4 ∅ ∈ {∅, 𝐴}
107, 9eqeltri 2821 . . 3 ∅ ∈ {∅, 𝐴}
116, 10eqeltrdi 2833 . 2 𝐴 ∈ V → {𝐴} ∈ {∅, 𝐴})
123, 11pm2.61i 182 1 {𝐴} ∈ {∅, 𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1533  wcel 2098  Vcvv 3466  c0 4315  {csn 4621  {cpr 4623   cuni 4900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-11 2146  ax-ext 2695  ax-nul 5297
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-sn 4622  df-pr 4624  df-uni 4901
This theorem is referenced by: (None)
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