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Theorem pwtrVD 39556
Description: Virtual deduction proof of pwtr 5068; see pwtrrVD 39557 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 4904 . . 3 (Tr 𝒫 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴))
2 idn1 39290 . . . . . . 7 (   Tr 𝐴   ▶   Tr 𝐴   )
3 idn2 39338 . . . . . . . . . 10 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   )
4 simpr 479 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑦 ∈ 𝒫 𝐴)
53, 4e2 39356 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑦 ∈ 𝒫 𝐴   )
6 elpwi 4310 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
75, 6e2 39356 . . . . . . . 8 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑦𝐴   )
8 simpl 474 . . . . . . . . 9 ((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧𝑦)
93, 8e2 39356 . . . . . . . 8 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑧𝑦   )
10 ssel 3736 . . . . . . . 8 (𝑦𝐴 → (𝑧𝑦𝑧𝐴))
117, 9, 10e22 39396 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑧𝐴   )
12 trss 4911 . . . . . . 7 (Tr 𝐴 → (𝑧𝐴𝑧𝐴))
132, 11, 12e12 39451 . . . . . 6 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑧𝐴   )
14 vex 3341 . . . . . . 7 𝑧 ∈ V
1514elpw 4306 . . . . . 6 (𝑧 ∈ 𝒫 𝐴𝑧𝐴)
1613, 15e2bir 39358 . . . . 5 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑧 ∈ 𝒫 𝐴   )
1716in2 39330 . . . 4 (   Tr 𝐴   ▶   ((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴)   )
1817gen12 39343 . . 3 (   Tr 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴)   )
19 biimpr 210 . . 3 ((Tr 𝒫 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴)) → (∀𝑧𝑦((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴) → Tr 𝒫 𝐴))
201, 18, 19e01 39416 . 2 (   Tr 𝐴   ▶   Tr 𝒫 𝐴   )
2120in1 39287 1 (Tr 𝐴 → Tr 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1628  wcel 2137  wss 3713  𝒫 cpw 4300  Tr wtr 4902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-v 3340  df-in 3720  df-ss 3727  df-pw 4302  df-uni 4587  df-tr 4903  df-vd1 39286  df-vd2 39294
This theorem is referenced by: (None)
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