Theorem snssg 4268

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 2676	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
2		sneq 4135	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
3	2	sseq1d 3595	. 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ⊆ 𝐵 ↔ {𝐴} ⊆ 𝐵))
4		vex 3176	. . 3 ⊢ 𝑥 ∈ V
5	4	snss 4259	. 2 ⊢ (𝑥 ∈ 𝐵 ↔ {𝑥} ⊆ 𝐵)
6	1, 3, 5	vtoclbg 3240	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977   ⊆ wss 3540  {csn 4125

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-sn 4126

This theorem is referenced by:  tppreqb  4277  snssi  4280  prssg  4290  fvimacnvALT  6244  fr3nr  6871  vdwapid1  15517  acsfn  16143  cycsubg2  17454  cycsubg2cl  17455  pgpfac1lem1  18296  pgpfac1lem3a  18298  pgpfac1lem3  18299  pgpfac1lem5  18301  pgpfaclem2  18304  lspsnid  18814  lidldvgen  19076  isneip  20719  elnei  20725  iscnp4  20877  cnpnei  20878  nlly2i  21089  1stckgenlem  21166  flimopn  21589  flimclslem  21598  fclsneii  21631  fcfnei  21649  limcvallem  23441  ellimc2  23447  limcflf  23451  limccnp  23461  limccnp2  23462  limcco  23463  lhop2  23582  plyrem  23864  isppw  24640  lpvtx  25734  h1did  27794  erdszelem8  30434  neibastop2  31526  prnc  33036  proot1mul  36796  uneqsn  37341  islptre  38686  rrxsnicc  39196