Theorem snelpw 4350

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 3848	. 2 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ⊆ 𝐵)
3	1	snex 4320	. . 3 ⊢ {𝐴} ∈ V
4	3	elpw 3694	. 2 ⊢ ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵)
5	2, 4	bitr4i 187	1 ⊢ (𝐴 ∈ 𝐵 ↔ {𝐴} ∈ 𝒫 𝐵)

Syntax hints:   ↔ wb 105   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220  𝒫 cpw 3688  {csn 3708

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714

This theorem is referenced by: (None)