Theorem snnex 7741

Description: The class of all singletons is a proper class. See also pwnex 7742. (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 7740	. . 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)
3	1, 2	sylibr 233	. 2 ⊢ (∀𝑦({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V)
4		vsnex 5428	. . 3 ⊢ {𝑦} ∈ V
5		vsnid 4664	. . 3 ⊢ 𝑦 ∈ {𝑦}
6	4, 5	pm3.2i 471	. 2 ⊢ ({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦})
7	3, 6	mpg 1799	1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V

Syntax hints:  ¬ wn 3   ∧ wa 396  ∀wal 1539   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2709   ∉ wnel 3046  Vcvv 3474  {csn 4627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2703  ax-sep 5298  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-un 3952  df-in 3954  df-ss 3964  df-sn 4628  df-pr 4630  df-uni 4908  df-iun 4998

This theorem is referenced by:  fiprc  9041