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Theorem snnex 7778
Description: The class of all singletons is a proper class. See also pwnex 7779. (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 7777 . . 3 (∀𝑦({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
2 df-nel 3047 . . 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 5434 . . 3 {𝑦} ∈ V
5 vsnid 4663 . . 3 𝑦 ∈ {𝑦}
64, 5pm3.2i 470 . 2 ({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦})
73, 6mpg 1797 1 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395  wal 1538   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wnel 3046  Vcvv 3480  {csn 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-un 3956  df-in 3958  df-ss 3968  df-sn 4627  df-pr 4629  df-uni 4908  df-iun 4993
This theorem is referenced by:  fiprc  9085
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