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Theorem unisn2 5312
Description: A version of unisn 4926 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 4925 . . 3 (𝐴 ∈ V → {𝐴} = 𝐴)
2 prid2g 4761 . . 3 (𝐴 ∈ V → 𝐴 ∈ {∅, 𝐴})
31, 2eqeltrd 2841 . 2 (𝐴 ∈ V → {𝐴} ∈ {∅, 𝐴})
4 snprc 4717 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
54biimpi 216 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
65unieqd 4920 . . 3 𝐴 ∈ V → {𝐴} = ∅)
7 uni0 4935 . . . 4 ∅ = ∅
8 0ex 5307 . . . . 5 ∅ ∈ V
98prid1 4762 . . . 4 ∅ ∈ {∅, 𝐴}
107, 9eqeltri 2837 . . 3 ∅ ∈ {∅, 𝐴}
116, 10eqeltrdi 2849 . 2 𝐴 ∈ V → {𝐴} ∈ {∅, 𝐴})
123, 11pm2.61i 182 1 {𝐴} ∈ {∅, 𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  {csn 4626  {cpr 4628   cuni 4907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-uni 4908
This theorem is referenced by: (None)
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