Mathbox for Alan Sare < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elpwgded Structured version   Visualization version   GIF version

Theorem elpwgded 40878
 Description: elpwgdedVD 41231 in conventional notation. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
elpwgded.1 (𝜑𝐴 ∈ V)
elpwgded.2 (𝜓𝐴𝐵)
Assertion
Ref Expression
elpwgded ((𝜑𝜓) → 𝐴 ∈ 𝒫 𝐵)

Proof of Theorem elpwgded
StepHypRef Expression
1 elpwgded.1 . 2 (𝜑𝐴 ∈ V)
2 elpwgded.2 . 2 (𝜓𝐴𝐵)
3 elpwg 4543 . . 3 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
43biimpar 480 . 2 ((𝐴 ∈ V ∧ 𝐴𝐵) → 𝐴 ∈ 𝒫 𝐵)
51, 2, 4syl2an 597 1 ((𝜑𝜓) → 𝐴 ∈ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 398   ∈ wcel 2107  Vcvv 3493   ⊆ wss 3934  𝒫 cpw 4537 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 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  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-in 3941  df-ss 3950  df-pw 4539 This theorem is referenced by:  sspwimp  41232  sspwimpALT  41239
 Copyright terms: Public domain W3C validator