Theorem snssg 3827

Description: The singleton formed on a set is included in a class if and only if the set is an element of that class. Theorem 7.4 of [Quine] p. 49. (Contributed by NM, 22-Jul-2001.) (Proof shortened by BJ, 1-Jan-2025.)

Assertion

Ref	Expression
snssg	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))

Proof of Theorem snssg

Step	Hyp	Ref	Expression
1		snssb 3826	. . 3 ⊢ ({𝐴} ⊆ 𝐵 ↔ (𝐴 ∈ V → 𝐴 ∈ 𝐵))
2	1	bicomi 132	. 2 ⊢ ((𝐴 ∈ V → 𝐴 ∈ 𝐵) ↔ {𝐴} ⊆ 𝐵)
3		elex 2824	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
4		imbibi 252	. 2 ⊢ (((𝐴 ∈ V → 𝐴 ∈ 𝐵) ↔ {𝐴} ⊆ 𝐵) → (𝐴 ∈ V → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)))
5	2, 3, 4	mpsyl 65	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 105   ∈ wcel 2203  Vcvv 2812   ⊆ wss 3210  {csn 3688

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-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 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2814  df-in 3216  df-ss 3223  df-sn 3694

This theorem is referenced by:  snss  3828  snssi  3837  snssd  3838  prssg  3850  snelpwg  4325  ordtri2orexmid  4644  ordtri2or2exmid  4692  ontri2orexmidim  4693  relsng  4852  fvimacnvi  5791  fvimacnv  5792  tpfidceq  7189  strslfv  13246  strslfv3  13247  imasaddfnlemg  13516  imasaddvallemg  13517  lspsnid  14542  psrplusgg  14820  isneip  14998  elnei  15004  iscnp4  15070  cnpnei  15071  lpvtx  16061