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Theorem unisn2 5312
Description: A version of unisn 4929 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 4928 . . 3 (𝐴 ∈ V → {𝐴} = 𝐴)
2 prid2g 4766 . . 3 (𝐴 ∈ V → 𝐴 ∈ {∅, 𝐴})
31, 2eqeltrd 2829 . 2 (𝐴 ∈ V → {𝐴} ∈ {∅, 𝐴})
4 snprc 4722 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
54biimpi 215 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
65unieqd 4921 . . 3 𝐴 ∈ V → {𝐴} = ∅)
7 uni0 4938 . . . 4 ∅ = ∅
8 0ex 5307 . . . . 5 ∅ ∈ V
98prid1 4767 . . . 4 ∅ ∈ {∅, 𝐴}
107, 9eqeltri 2825 . . 3 ∅ ∈ {∅, 𝐴}
116, 10eqeltrdi 2837 . 2 𝐴 ∈ V → {𝐴} ∈ {∅, 𝐴})
123, 11pm2.61i 182 1 {𝐴} ∈ {∅, 𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1534  wcel 2099  Vcvv 3471  c0 4323  {csn 4629  {cpr 4631   cuni 4908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-11 2147  ax-ext 2699  ax-nul 5306
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-sn 4630  df-pr 4632  df-uni 4909
This theorem is referenced by: (None)
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