Theorem snprc 3703

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 3655	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2	1	exbii 1629	. . 3 ⊢ (∃𝑥 𝑥 ∈ {𝐴} ↔ ∃𝑥 𝑥 = 𝐴)
3	2	notbii 670	. 2 ⊢ (¬ ∃𝑥 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 = 𝐴)
4		eq0 3483	. . 3 ⊢ ({𝐴} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ {𝐴})
5		alnex 1523	. . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴})
6	4, 5	bitri 184	. 2 ⊢ ({𝐴} = ∅ ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴})
7		isset 2780	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
8	7	notbii 670	. 2 ⊢ (¬ 𝐴 ∈ V ↔ ¬ ∃𝑥 𝑥 = 𝐴)
9	3, 6, 8	3bitr4ri 213	1 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)

Syntax hints:  ¬ wn 3   ↔ wb 105  ∀wal 1371   = wceq 1373  ∃wex 1516   ∈ wcel 2177  Vcvv 2773  ∅c0 3464  {csn 3638

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-dif 3172  df-nul 3465  df-sn 3644

This theorem is referenced by:  prprc1  3746  prprc  3748  snexprc  4238  sucprc  4467  snnen2oprc  6972  unsnfidcex  7032