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

Theorem snnex 7777
Description: The class of all singletons is a proper class. See also pwnex 7778. (Contributed by NM, 10-Oct-2008.) (Proof shortened by Eric Schmidt, 7-Dec-2008.) (Proof shortened by BJ, 5-Dec-2021.)
Assertion
Ref Expression
snnex {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
Distinct variable group:   𝑥,𝑦

Proof of Theorem snnex
StepHypRef Expression
1 abnex 7776 . . 3 (∀𝑦({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
2 df-nel 3045 . . 3 ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
31, 2sylibr 234 . 2 (∀𝑦({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V)
4 vsnex 5440 . . 3 {𝑦} ∈ V
5 vsnid 4668 . . 3 𝑦 ∈ {𝑦}
64, 5pm3.2i 470 . 2 ({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦})
73, 6mpg 1794 1 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wal 1535   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wnel 3044  Vcvv 3478  {csn 4631
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-sep 5302  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-un 3968  df-in 3970  df-ss 3980  df-sn 4632  df-pr 4634  df-uni 4913  df-iun 4998
This theorem is referenced by:  fiprc  9084
  Copyright terms: Public domain W3C validator