Theorem snnex 7703

Description: The class of all singletons is a proper class. See also pwnex 7704. (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 7702	. . 3 ⊢ (∀𝑦({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦}) → ¬ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∈ V)
2		df-nel 3037	. . 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 5379	. . 3 ⊢ {𝑦} ∈ V
5		vsnid 4620	. . 3 ⊢ 𝑦 ∈ {𝑦}
6	4, 5	pm3.2i 470	. 2 ⊢ ({𝑦} ∈ V ∧ 𝑦 ∈ {𝑦})
7	3, 6	mpg 1798	1 ⊢ {𝑥 ∣ ∃𝑦 𝑥 = {𝑦}} ∉ V

Syntax hints:  ¬ wn 3   ∧ wa 395  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2113  {cab 2714   ∉ wnel 3036  Vcvv 3440  {csn 4580

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2708  ax-sep 5241  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-un 3906  df-in 3908  df-ss 3918  df-sn 4581  df-pr 4583  df-uni 4864  df-iun 4948

This theorem is referenced by:  fiprc  8981  termcnex  49821