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

Theorem pwtrrVD 41036
Description: Virtual deduction proof of pwtr 5336; see pwtrVD 41035 for the converse. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
pwtrrVD.1 𝐴 ∈ V
Assertion
Ref Expression
pwtrrVD (Tr 𝒫 𝐴 → Tr 𝐴)

Proof of Theorem pwtrrVD
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 5165 . . 3 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
2 idn1 40785 . . . . . . . 8 (   Tr 𝒫 𝐴   ▶   Tr 𝒫 𝐴   )
3 idn2 40824 . . . . . . . . 9 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   (𝑧𝑦𝑦𝐴)   )
4 simpr 485 . . . . . . . . 9 ((𝑧𝑦𝑦𝐴) → 𝑦𝐴)
53, 4e2 40842 . . . . . . . 8 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦𝐴   )
6 pwtrrVD.1 . . . . . . . . 9 𝐴 ∈ V
76pwid 4554 . . . . . . . 8 𝐴 ∈ 𝒫 𝐴
8 trel 5170 . . . . . . . . 9 (Tr 𝒫 𝐴 → ((𝑦𝐴𝐴 ∈ 𝒫 𝐴) → 𝑦 ∈ 𝒫 𝐴))
98expd 416 . . . . . . . 8 (Tr 𝒫 𝐴 → (𝑦𝐴 → (𝐴 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴)))
102, 5, 7, 9e120 40874 . . . . . . 7 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦 ∈ 𝒫 𝐴   )
11 elpwi 4547 . . . . . . 7 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
1210, 11e2 40842 . . . . . 6 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦𝐴   )
13 simpl 483 . . . . . . 7 ((𝑧𝑦𝑦𝐴) → 𝑧𝑦)
143, 13e2 40842 . . . . . 6 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑧𝑦   )
15 ssel 3958 . . . . . 6 (𝑦𝐴 → (𝑧𝑦𝑧𝐴))
1612, 14, 15e22 40882 . . . . 5 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑧𝐴   )
1716in2 40816 . . . 4 (   Tr 𝒫 𝐴   ▶   ((𝑧𝑦𝑦𝐴) → 𝑧𝐴)   )
1817gen12 40829 . . 3 (   Tr 𝒫 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)   )
19 biimpr 221 . . 3 ((Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)) → (∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴) → Tr 𝐴))
201, 18, 19e01 40902 . 2 (   Tr 𝒫 𝐴   ▶   Tr 𝐴   )
2120in1 40782 1 (Tr 𝒫 𝐴 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1526  wcel 2105  Vcvv 3492  wss 3933  𝒫 cpw 4535  Tr wtr 5163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-in 3940  df-ss 3949  df-pw 4537  df-uni 4831  df-tr 5164  df-vd1 40781  df-vd2 40789
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator