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

Step	Hyp	Ref	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)
3	1, 2	sylibr 234	. 2 ⊢ (∀𝑦({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) → {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V)
4		vsnex 5434	. . 3 ⊢ {𝑦} ∈ V
5		vsnid 4663	. . 3 ⊢ 𝑦 ∈ {𝑦}
6	4, 5	pm3.2i 470	. 2 ⊢ ({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦})
7	3, 6	mpg 1797	1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V

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