Theorem snssg 3522

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 2141	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2		sneq 3409	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
3	2	sseq1d 3026	. 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ⊆ 𝐵 ↔ {𝐴} ⊆ 𝐵))
4		vex 2604	. . 3 ⊢ 𝑥 ∈ V
5	4	snss 3516	. 2 ⊢ (𝑥 ∈ 𝐵 ↔ {𝑥} ⊆ 𝐵)
6	1, 3, 5	vtoclbg 2659	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 103   = wceq 1284   ∈ wcel 1433   ⊆ wss 2973  {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-sn 3404

This theorem is referenced by:  snssi  3529  snssd  3530  prssg  3542  ordtri2orexmid  4266  ordtri2or2exmid  4314  fvimacnvi  5302  fvimacnv  5303