Theorem snprc 3620

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 3573	. . . 4 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
2	1	exbii 1582	. . 3 ⊢ (∃𝑥 𝑥 ∈ {𝐴} ↔ ∃𝑥 𝑥 = 𝐴)
3	2	notbii 658	. 2 ⊢ (¬ ∃𝑥 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 = 𝐴)
4		eq0 3408	. . 3 ⊢ ({𝐴} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ {𝐴})
5		alnex 1476	. . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ {𝐴} ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴})
6	4, 5	bitri 183	. 2 ⊢ ({𝐴} = ∅ ↔ ¬ ∃𝑥 𝑥 ∈ {𝐴})
7		isset 2715	. . 3 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
8	7	notbii 658	. 2 ⊢ (¬ 𝐴 ∈ V ↔ ¬ ∃𝑥 𝑥 = 𝐴)
9	3, 6, 8	3bitr4ri 212	1 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)

Syntax hints:  ¬ wn 3   ↔ wb 104  ∀wal 1330   = wceq 1332  ∃wex 1469   ∈ wcel 2125  Vcvv 2709  ∅c0 3390  {csn 3556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-ext 2136

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-v 2711  df-dif 3100  df-nul 3391  df-sn 3562

This theorem is referenced by:  prprc1  3663  prprc  3665  snexprc  4142  sucprc  4367  snnen2oprc  6794  unsnfidcex  6853