MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unisn2 Structured version   Visualization version   GIF version

Theorem unisn2 5318
Description: A version of unisn 4931 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 4930 . . 3 (𝐴 ∈ V → {𝐴} = 𝐴)
2 prid2g 4766 . . 3 (𝐴 ∈ V → 𝐴 ∈ {∅, 𝐴})
31, 2eqeltrd 2839 . 2 (𝐴 ∈ V → {𝐴} ∈ {∅, 𝐴})
4 snprc 4722 . . . . 5 𝐴 ∈ V ↔ {𝐴} = ∅)
54biimpi 216 . . . 4 𝐴 ∈ V → {𝐴} = ∅)
65unieqd 4925 . . 3 𝐴 ∈ V → {𝐴} = ∅)
7 uni0 4940 . . . 4 ∅ = ∅
8 0ex 5313 . . . . 5 ∅ ∈ V
98prid1 4767 . . . 4 ∅ ∈ {∅, 𝐴}
107, 9eqeltri 2835 . . 3 ∅ ∈ {∅, 𝐴}
116, 10eqeltrdi 2847 . 2 𝐴 ∈ V → {𝐴} ∈ {∅, 𝐴})
123, 11pm2.61i 182 1 {𝐴} ∈ {∅, 𝐴}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  {csn 4631  {cpr 4633   cuni 4912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-uni 4913
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator