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Theorem snelpwg 5446
Description: A singleton of a set is a member of the powerclass of a class if and only if that set is a member of that class. (Contributed by NM, 1-Apr-1998.) Put in closed form and avoid ax-nul 5305. (Revised by BJ, 17-Jan-2025.)
Assertion
Ref Expression
snelpwg (𝐴𝑉 → (𝐴𝐵 ↔ {𝐴} ∈ 𝒫 𝐵))

Proof of Theorem snelpwg
StepHypRef Expression
1 snssg 4782 . 2 (𝐴𝑉 → (𝐴𝐵 ↔ {𝐴} ⊆ 𝐵))
2 snexg 5434 . . 3 (𝐴𝑉 → {𝐴} ∈ V)
3 elpwg 4602 . . 3 ({𝐴} ∈ V → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵))
42, 3syl 17 . 2 (𝐴𝑉 → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵))
51, 4bitr4d 282 1 (𝐴𝑉 → (𝐴𝐵 ↔ {𝐴} ∈ 𝒫 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wcel 2107  Vcvv 3479  wss 3950  𝒫 cpw 4599  {csn 4625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-un 3955  df-ss 3967  df-pw 4601  df-sn 4626  df-pr 4628
This theorem is referenced by:  snelpwi  5447  snelpw  5449
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