Theorem snssg 3728

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 3727	. . 3 ⊢ ({𝐴} ⊆ 𝐵 ↔ (𝐴 ∈ V → 𝐴 ∈ 𝐵))
2	1	bicomi 132	. 2 ⊢ ((𝐴 ∈ V → 𝐴 ∈ 𝐵) ↔ {𝐴} ⊆ 𝐵)
3		elex 2750	. 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 2148  Vcvv 2739   ⊆ wss 3131  {csn 3594

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-ss 3144  df-sn 3600

This theorem is referenced by:  snss  3729  snssi  3738  snssd  3739  prssg  3751  ordtri2orexmid  4524  ordtri2or2exmid  4572  ontri2orexmidim  4573  relsng  4731  fvimacnvi  5632  fvimacnv  5633  strslfv  12509  imasaddfnlemg  12740  imasaddvallemg  12741  lspsnid  13498  isneip  13731  elnei  13737  iscnp4  13803  cnpnei  13804