Theorem snnex 7737

Description: The class of all singletons is a proper class. See also pwnex 7738. (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 7736	. . 3 ⊢ (∀𝑦({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
2		df-nel 3031	. . 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 5392	. . 3 ⊢ {𝑦} ∈ V
5		vsnid 4630	. . 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 2109  {cab 2708   ∉ wnel 3030  Vcvv 3450  {csn 4592

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 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2702  ax-sep 5254  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nel 3031  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-un 3922  df-in 3924  df-ss 3934  df-sn 4593  df-pr 4595  df-uni 4875  df-iun 4960

This theorem is referenced by:  fiprc  9019  termcnex  49569