Theorem snelpwiOLD 5440

Description: Obsolete version of snelpwi 5439 as of 17-Jan-2025. (Contributed by NM, 28-May-1995.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
snelpwiOLD	⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵)

Proof of Theorem snelpwiOLD

Step	Hyp	Ref	Expression
1		snssi 4807	. 2 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵)
2		snex 5427	. . 3 ⊢ {𝐴} ∈ V
3	2	elpw 4602	. 2 ⊢ ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵)
4	1, 3	sylibr 233	1 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ∈ 𝒫 𝐵)

Syntax hints:   → wi 4   ∈ wcel 2098   ⊆ wss 3939  𝒫 cpw 4598  {csn 4624

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3465  df-dif 3942  df-un 3944  df-ss 3956  df-nul 4319  df-pw 4600  df-sn 4625  df-pr 4627

This theorem is referenced by: (None)