Theorem snss 3839

Description: The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 5-Aug-1993.)

Hypothesis

Ref	Expression
snss.1	⊢ A ∈ V

Assertion

Ref	Expression
snss	⊢ (A ∈ B ↔ {A} ⊆ B)

Proof of Theorem snss

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elsn 3749	. . . 4 ⊢ (x ∈ {A} ↔ x = A)
2	1	imbi1i 315	. . 3 ⊢ ((x ∈ {A} → x ∈ B) ↔ (x = A → x ∈ B))
3	2	albii 1566	. 2 ⊢ (∀x(x ∈ {A} → x ∈ B) ↔ ∀x(x = A → x ∈ B))
4		dfss2 3263	. 2 ⊢ ({A} ⊆ B ↔ ∀x(x ∈ {A} → x ∈ B))
5		snss.1	. . 3 ⊢ A ∈ V
6	5	clel2 2976	. 2 ⊢ (A ∈ B ↔ ∀x(x = A → x ∈ B))
7	3, 4, 6	3bitr4ri 269	1 ⊢ (A ∈ B ↔ {A} ⊆ B)

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  Vcvv 2860   ⊆ wss 3258  {csn 3738

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-sn 3742

This theorem is referenced by:  snssg  3845  prss  3862  tpss  3872  sspwb  4119  elpw1  4145  nnsucelrlem3  4427  nnsucelr  4429  tfinnn  4535  vfinspeqtncv  4554  brssetsn  4760  fvimacnvi  5403  fvimacnv  5404  fnressn  5439  dfnnc3  5886  mapsn  6027  nc0le1  6217  frecxp  6315