Theorem snelpw 4243

Description: A singleton of a set belongs to the power class of a class containing the set. (Contributed by NM, 1-Apr-1998.)

Hypothesis

Ref	Expression
snelpw.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
snelpw	⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)

Proof of Theorem snelpw

Step	Hyp	Ref	Expression
1		snelpw.1	. . 3 ⊢ 𝐴 ∈ V
2	1	snss 3754	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)
3	1	snex 4215	. . 3 ⊢ {𝐴} ∈ V
4	3	elpw 3608	. 2 ⊢ ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵)
5	2, 4	bitr4i 187	1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)

Syntax hints:   ↔ wb 105   ∈ wcel 2164  Vcvv 2760   ⊆ wss 3154  𝒫 cpw 3602  {csn 3619

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625

This theorem is referenced by: (None)