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

Theorem snprc 4285
 Description: The singleton of a proper class (one that doesn't exist) is the empty set. Theorem 7.2 of [Quine] p. 48. (Contributed by NM, 21-Jun-1993.)
Assertion
Ref Expression
snprc 𝐴 ∈ V ↔ {𝐴} = ∅)

Proof of Theorem snprc
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 velsn 4226 . . . 4 (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
21exbii 1814 . . 3 (∃𝑥 𝑥 ∈ {𝐴} ↔ ∃𝑥 𝑥 = 𝐴)
3 neq0 3963 . . 3 (¬ {𝐴} = ∅ ↔ ∃𝑥 𝑥 ∈ {𝐴})
4 isset 3238 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
52, 3, 43bitr4i 292 . 2 (¬ {𝐴} = ∅ ↔ 𝐴 ∈ V)
65con1bii 345 1 𝐴 ∈ V ↔ {𝐴} = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196   = wceq 1523  ∃wex 1744   ∈ wcel 2030  Vcvv 3231  ∅c0 3948  {csn 4210 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-nul 3949  df-sn 4211 This theorem is referenced by:  snnzb  4286  rabsnif  4290  prprc1  4332  prprc  4334  unisn2  4827  snexALT  4882  snex  4938  posn  5221  frsn  5223  relimasn  5523  elimasni  5527  inisegn0  5532  dmsnsnsn  5649  sucprc  5838  dffv3  6225  fconst5  6512  1stval  7212  2ndval  7213  ecexr  7792  snfi  8079  domunsn  8151  snnen2o  8190  hashrabrsn  13199  hashrabsn01  13200  hashrabsn1  13201  elprchashprn2  13222  hashsn01  13242  hash2pwpr  13296  efgrelexlema  18208  usgr1v  26193  1conngr  27172  frgr1v  27251  n0lplig  27465  eldm3  31777  opelco3  31802  fvsingle  32152  unisnif  32157  funpartlem  32174  bj-sngltag  33096  bj-restsnid  33165  bj-snmoore  33193  wopprc  37914  uneqsn  38638
 Copyright terms: Public domain W3C validator