Theorem snss 1849

Description: The singleton of an element of a class is a subset of the class. Theorem 7.4 of [Quine] p. 49.

Hypothesis

Ref	Expression
snss.1	⊢ A ∈ V

Assertion

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

Proof of Theorem snss

Step	Hyp	Ref	Expression
1		elsn 1820	. . . 4 ⊢ (x ∈ {A} ↔ x = A)
2	1	imbi1i 161	. . 3 ⊢ ((x ∈ {A} → x ∈ B) ↔ (x = A → x ∈ B))
3	2	bial 695	. 2 ⊢ (∀x(x ∈ {A} → x ∈ B) ↔ ∀x(x = A → x ∈ B))
4		dfss2 1497	. 2 ⊢ ({A} ⊆ B ↔ ∀x(x ∈ {A} → x ∈ B))
5		snss.1	. . 3 ⊢ A ∈ V
6	5	clel2 1374	. 2 ⊢ (A ∈ B ↔ ∀x(x = A → x ∈ B))
7	3, 4, 6	3bitr4r 159	1 ⊢ (A ∈ B ↔ {A} ⊆ B)

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  {csn 1808

This theorem is referenced by:  snssg 1850  snelpw 1861  sspwb 1863  nnullss 1880  exss 1881  pwpw0 1883  pwssun 1917  frirr 2176  fnressn 2897  xpdom3 3347  limensuci 3401  zfregs 3491  nn0ssz 4578  spansn 5462

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-in 1491  df-ss 1492  df-sn 1811

metamath.org