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Theorem unisn2 4785
Description: A version of unisn 4442 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 4443 . . 3 (𝐴 ∈ V → {𝐴} = 𝐴)
2 prid2g 4287 . . 3 (𝐴 ∈ V → 𝐴 ∈ {∅, 𝐴})
31, 2eqeltrd 2699 . 2 (𝐴 ∈ V → {𝐴} ∈ {∅, 𝐴})
4 snprc 4244 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
54biimpi 206 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
65unieqd 4437 . . 3 𝐴 ∈ V → {𝐴} = ∅)
7 uni0 4456 . . . 4 ∅ = ∅
8 0ex 4781 . . . . 5 ∅ ∈ V
98prid1 4288 . . . 4 ∅ ∈ {∅, 𝐴}
107, 9eqeltri 2695 . . 3 ∅ ∈ {∅, 𝐴}
116, 10syl6eqel 2707 . 2 𝐴 ∈ V → {𝐴} ∈ {∅, 𝐴})
123, 11pm2.61i 176 1 {𝐴} ∈ {∅, 𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1481  wcel 1988  Vcvv 3195  c0 3907  {csn 4168  {cpr 4170   cuni 4427
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-nul 4780
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-sn 4169  df-pr 4171  df-uni 4428
This theorem is referenced by: (None)
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