Theorem snnex 7745

Description: The class of all singletons is a proper class. See also pwnex 7746. (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

Step	Hyp	Ref	Expression
1		abnex 7744	. . 3 ⊢ (∀𝑦({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
2		df-nel 3048	. . 3 ⊢ ({𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V ↔ ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
3	1, 2	sylibr 233	. 2 ⊢ (∀𝑦({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V)
4		vsnex 5430	. . 3 ⊢ {𝑦} ∈ V
5		vsnid 4666	. . 3 ⊢ 𝑦 ∈ {𝑦}
6	4, 5	pm3.2i 472	. 2 ⊢ ({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦})
7	3, 6	mpg 1800	1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V

Syntax hints:  ¬ wn 3   ∧ wa 397  ∀wal 1540   = wceq 1542  ∃wex 1782   ∈ wcel 2107  {cab 2710   ∉ wnel 3047  Vcvv 3475  {csn 4629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-11 2155  ax-ext 2704  ax-sep 5300  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nel 3048  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-un 3954  df-in 3956  df-ss 3966  df-sn 4630  df-pr 4632  df-uni 4910  df-iun 5000

This theorem is referenced by:  fiprc  9045