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Theorem snnex 7164
Description: The class of all singletons is a proper class. See also pwnex 7166. (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 7163 . . 3 (∀𝑦({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
2 df-nel 3041 . . 3 ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
31, 2sylibr 225 . 2 (∀𝑦({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V)
4 snex 5064 . . 3 {𝑦} ∈ V
5 vsnid 4367 . . 3 𝑦 ∈ {𝑦}
64, 5pm3.2i 462 . 2 ({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦})
73, 6mpg 1892 1 {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384  wal 1650   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wnel 3040  Vcvv 3350  {csn 4334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-nel 3041  df-ral 3060  df-rex 3061  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-sn 4335  df-pr 4337  df-uni 4595  df-iun 4678
This theorem is referenced by:  fiprc  8246
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