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Theorem unisn2 4722
 Description: A version of unisn 4387 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 4388 . . 3 (𝐴 ∈ V → {𝐴} = 𝐴)
2 prid2g 4240 . . 3 (𝐴 ∈ V → 𝐴 ∈ {∅, 𝐴})
31, 2eqeltrd 2688 . 2 (𝐴 ∈ V → {𝐴} ∈ {∅, 𝐴})
4 snprc 4197 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
54biimpi 205 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
65unieqd 4382 . . 3 𝐴 ∈ V → {𝐴} = ∅)
7 uni0 4401 . . . 4 ∅ = ∅
8 0ex 4718 . . . . 5 ∅ ∈ V
98prid1 4241 . . . 4 ∅ ∈ {∅, 𝐴}
107, 9eqeltri 2684 . . 3 ∅ ∈ {∅, 𝐴}
116, 10syl6eqel 2696 . 2 𝐴 ∈ V → {𝐴} ∈ {∅, 𝐴})
123, 11pm2.61i 175 1 {𝐴} ∈ {∅, 𝐴}
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  {csn 4125  {cpr 4127  ∪ cuni 4372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373 This theorem is referenced by: (None)
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