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Theorem snelpwrVD 41145
 Description: Virtual deduction proof of snelpwi 5327. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
snelpwrVD (𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)

Proof of Theorem snelpwrVD
StepHypRef Expression
1 snex 5322 . . 3 {𝐴} ∈ V
2 idn1 40888 . . . 4 (   𝐴𝐵   ▶   𝐴𝐵   )
3 snssi 4733 . . . 4 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
42, 3e1a 40941 . . 3 (   𝐴𝐵   ▶   {𝐴} ⊆ 𝐵   )
5 elpwg 4543 . . . 4 ({𝐴} ∈ V → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵))
65biimprd 250 . . 3 ({𝐴} ∈ V → ({𝐴} ⊆ 𝐵 → {𝐴} ∈ 𝒫 𝐵))
71, 4, 6e01 41005 . 2 (   𝐴𝐵   ▶   {𝐴} ∈ 𝒫 𝐵   )
87in1 40885 1 (𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2107  Vcvv 3493   ⊆ wss 3934  𝒫 cpw 4537  {csn 4559 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-pw 4539  df-sn 4560  df-pr 4562  df-vd1 40884 This theorem is referenced by: (None)
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