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Theorem unisn2 5273
Description: A version of unisn 4891 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 4890 . . 3 (𝐴 ∈ V → {𝐴} = 𝐴)
2 prid2g 4726 . . 3 (𝐴 ∈ V → 𝐴 ∈ {∅, 𝐴})
31, 2eqeltrd 2834 . 2 (𝐴 ∈ V → {𝐴} ∈ {∅, 𝐴})
4 snprc 4682 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
54biimpi 215 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
65unieqd 4883 . . 3 𝐴 ∈ V → {𝐴} = ∅)
7 uni0 4900 . . . 4 ∅ = ∅
8 0ex 5268 . . . . 5 ∅ ∈ V
98prid1 4727 . . . 4 ∅ ∈ {∅, 𝐴}
107, 9eqeltri 2830 . . 3 ∅ ∈ {∅, 𝐴}
116, 10eqeltrdi 2842 . 2 𝐴 ∈ V → {𝐴} ∈ {∅, 𝐴})
123, 11pm2.61i 182 1 {𝐴} ∈ {∅, 𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  Vcvv 3447  c0 4286  {csn 4590  {cpr 4592   cuni 4869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-nul 5267
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-sn 4591  df-pr 4593  df-uni 4870
This theorem is referenced by: (None)
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