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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 (𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)

Proof of Theorem snelpwiOLD
StepHypRef Expression
1 snssi 4807 . 2 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
2 snex 5427 . . 3 {𝐴} ∈ V
32elpw 4602 . 2 ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵)
41, 3sylibr 233 1 (𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
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)
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