Theorem snprc 3738

Description: The singleton of a proper class (one that doesn't exist) is the empty set. Theorem 7.2 of [Quine] p. 48. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
snprc	⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)

Proof of Theorem snprc

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		velsn 3690	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2	1	exbii 1654	. . 3 ⊢ (∃𝑥 𝑥 ∈ {𝐴} ↔ ∃𝑥 𝑥 = 𝐴)
3	2	notbii 674	. 2 ⊢ (¬ ∃𝑥 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 = 𝐴)
4		eq0 3515	. . 3 ⊢ ({𝐴} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ {𝐴})
5		alnex 1548	. . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴})
6	4, 5	bitri 184	. 2 ⊢ ({𝐴} = ∅ ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴})
7		isset 2810	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
8	7	notbii 674	. 2 ⊢ (¬ 𝐴 ∈ V ↔ ¬ ∃𝑥 𝑥 = 𝐴)
9	3, 6, 8	3bitr4ri 213	1 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)

Syntax hints:  ¬ wn 3   ↔ wb 105  ∀wal 1396   = wceq 1398  ∃wex 1541   ∈ wcel 2202  Vcvv 2803  ∅c0 3496  {csn 3673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-nul 3497  df-sn 3679

This theorem is referenced by:  prprc1  3784  prprc  3786  snexprc  4282  sucprc  4515  snnen2oprc  7089  unsnfidcex  7155