Theorem snssg 3656

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 2202	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2		sneq 3538	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
3	2	sseq1d 3126	. 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ⊆ 𝐵 ↔ {𝐴} ⊆ 𝐵))
4		vex 2689	. . 3 ⊢ 𝑥 ∈ V
5	4	snss 3649	. 2 ⊢ (𝑥 ∈ 𝐵 ↔ {𝑥} ⊆ 𝐵)
6	1, 3, 5	vtoclbg 2747	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1331   ∈ wcel 1480   ⊆ wss 3071  {csn 3527

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-ss 3084  df-sn 3533

This theorem is referenced by:  snssi  3664  snssd  3665  prssg  3677  ordtri2orexmid  4438  ordtri2or2exmid  4486  relsng  4642  fvimacnvi  5534  fvimacnv  5535  strslfv  12003  isneip  12315  elnei  12321  iscnp4  12387  cnpnei  12388