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Theorem snelpwrVD 39380
Description: Virtual deduction proof of snelpwi 4942. (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 4938 . . 3 {𝐴} ∈ V
2 idn1 39107 . . . 4 (   𝐴𝐵   ▶   𝐴𝐵   )
3 snssi 4371 . . . 4 (𝐴𝐵 → {𝐴} ⊆ 𝐵)
42, 3e1a 39169 . . 3 (   𝐴𝐵   ▶   {𝐴} ⊆ 𝐵   )
5 elpwg 4199 . . . 4 ({𝐴} ∈ V → ({𝐴} ∈ 𝒫 𝐵 ↔ {𝐴} ⊆ 𝐵))
65biimprd 238 . . 3 ({𝐴} ∈ V → ({𝐴} ⊆ 𝐵 → {𝐴} ∈ 𝒫 𝐵))
71, 4, 6e01 39233 . 2 (   𝐴𝐵   ▶   {𝐴} ∈ 𝒫 𝐵   )
87in1 39104 1 (𝐴𝐵 → {𝐴} ∈ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2030  Vcvv 3231  wss 3607  𝒫 cpw 4191  {csn 4210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-sn 4211  df-pr 4213  df-vd1 39103
This theorem is referenced by: (None)
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