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Theorem pwtrVD 39820
Description: Virtual deduction proof of pwtr 5112; see pwtrrVD 39821 for the converse. (Contributed by Alan Sare, 25-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
pwtrVD (Tr 𝐴 → Tr 𝒫 𝐴)

Proof of Theorem pwtrVD
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 4947 . . 3 (Tr 𝒫 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴))
2 idn1 39560 . . . . . . 7 (   Tr 𝐴   ▶   Tr 𝐴   )
3 idn2 39608 . . . . . . . . . 10 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   )
4 simpr 478 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑦 ∈ 𝒫 𝐴)
53, 4e2 39626 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑦 ∈ 𝒫 𝐴   )
6 elpwi 4359 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
75, 6e2 39626 . . . . . . . 8 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑦𝐴   )
8 simpl 475 . . . . . . . . 9 ((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧𝑦)
93, 8e2 39626 . . . . . . . 8 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑧𝑦   )
10 ssel 3792 . . . . . . . 8 (𝑦𝐴 → (𝑧𝑦𝑧𝐴))
117, 9, 10e22 39666 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑧𝐴   )
12 trss 4954 . . . . . . 7 (Tr 𝐴 → (𝑧𝐴𝑧𝐴))
132, 11, 12e12 39720 . . . . . 6 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑧𝐴   )
14 vex 3388 . . . . . . 7 𝑧 ∈ V
1514elpw 4355 . . . . . 6 (𝑧 ∈ 𝒫 𝐴𝑧𝐴)
1613, 15e2bir 39628 . . . . 5 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑧 ∈ 𝒫 𝐴   )
1716in2 39600 . . . 4 (   Tr 𝐴   ▶   ((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴)   )
1817gen12 39613 . . 3 (   Tr 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴)   )
19 biimpr 212 . . 3 ((Tr 𝒫 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴)) → (∀𝑧𝑦((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴) → Tr 𝒫 𝐴))
201, 18, 19e01 39686 . 2 (   Tr 𝐴   ▶   Tr 𝒫 𝐴   )
2120in1 39557 1 (Tr 𝐴 → Tr 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385  wal 1651  wcel 2157  wss 3769  𝒫 cpw 4349  Tr wtr 4945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-v 3387  df-in 3776  df-ss 3783  df-pw 4351  df-uni 4629  df-tr 4946  df-vd1 39556  df-vd2 39564
This theorem is referenced by: (None)
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