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Theorem pwtrrVD 43634
Description: Virtual deduction proof of pwtr 5453; see pwtrVD 43633 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 5268 . . 3 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
2 idn1 43383 . . . . . . . 8 (   Tr 𝒫 𝐴   ▶   Tr 𝒫 𝐴   )
3 idn2 43422 . . . . . . . . 9 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   (𝑧𝑦𝑦𝐴)   )
4 simpr 486 . . . . . . . . 9 ((𝑧𝑦𝑦𝐴) → 𝑦𝐴)
53, 4e2 43440 . . . . . . . 8 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦𝐴   )
6 pwtrrVD.1 . . . . . . . . 9 𝐴 ∈ V
76pwid 4625 . . . . . . . 8 𝐴 ∈ 𝒫 𝐴
8 trel 5275 . . . . . . . . 9 (Tr 𝒫 𝐴 → ((𝑦𝐴𝐴 ∈ 𝒫 𝐴) → 𝑦 ∈ 𝒫 𝐴))
98expd 417 . . . . . . . 8 (Tr 𝒫 𝐴 → (𝑦𝐴 → (𝐴 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴)))
102, 5, 7, 9e120 43472 . . . . . . 7 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦 ∈ 𝒫 𝐴   )
11 elpwi 4610 . . . . . . 7 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
1210, 11e2 43440 . . . . . 6 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦𝐴   )
13 simpl 484 . . . . . . 7 ((𝑧𝑦𝑦𝐴) → 𝑧𝑦)
143, 13e2 43440 . . . . . 6 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑧𝑦   )
15 ssel 3976 . . . . . 6 (𝑦𝐴 → (𝑧𝑦𝑧𝐴))
1612, 14, 15e22 43480 . . . . 5 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑧𝐴   )
1716in2 43414 . . . 4 (   Tr 𝒫 𝐴   ▶   ((𝑧𝑦𝑦𝐴) → 𝑧𝐴)   )
1817gen12 43427 . . 3 (   Tr 𝒫 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)   )
19 biimpr 219 . . 3 ((Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)) → (∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴) → Tr 𝐴))
201, 18, 19e01 43500 . 2 (   Tr 𝒫 𝐴   ▶   Tr 𝐴   )
2120in1 43380 1 (Tr 𝒫 𝐴 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  wcel 2107  Vcvv 3475  wss 3949  𝒫 cpw 4603  Tr wtr 5266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-in 3956  df-ss 3966  df-pw 4605  df-uni 4910  df-tr 5267  df-vd1 43379  df-vd2 43387
This theorem is referenced by: (None)
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