Theorem snssg 3709

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, 22-Jul-2001.)

Assertion

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

Proof of Theorem snssg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2229	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2		sneq 3587	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
3	2	sseq1d 3171	. 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ⊆ 𝐵 ↔ {𝐴} ⊆ 𝐵))
4		vex 2729	. . 3 ⊢ 𝑥 ∈ V
5	4	snss 3702	. 2 ⊢ (𝑥 ∈ 𝐵 ↔ {𝑥} ⊆ 𝐵)
6	1, 3, 5	vtoclbg 2787	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1343   ∈ wcel 2136   ⊆ wss 3116  {csn 3576

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-sn 3582

This theorem is referenced by:  snssi  3717  snssd  3718  prssg  3730  ordtri2orexmid  4500  ordtri2or2exmid  4548  ontri2orexmidim  4549  relsng  4707  fvimacnvi  5599  fvimacnv  5600  strslfv  12438  isneip  12786  elnei  12792  iscnp4  12858  cnpnei  12859