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Theorem pwtrVD 38542
 Description: Virtual deduction proof of pwtr 4882; see pwtrrVD 38543 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 4714 . . 3 (Tr 𝒫 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴))
2 idn1 38272 . . . . . . 7 (   Tr 𝐴   ▶   Tr 𝐴   )
3 idn2 38320 . . . . . . . . . 10 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   )
4 simpr 477 . . . . . . . . . 10 ((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑦 ∈ 𝒫 𝐴)
53, 4e2 38338 . . . . . . . . 9 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑦 ∈ 𝒫 𝐴   )
6 elpwi 4140 . . . . . . . . 9 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
75, 6e2 38338 . . . . . . . 8 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑦𝐴   )
8 simpl 473 . . . . . . . . 9 ((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧𝑦)
93, 8e2 38338 . . . . . . . 8 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑧𝑦   )
10 ssel 3577 . . . . . . . 8 (𝑦𝐴 → (𝑧𝑦𝑧𝐴))
117, 9, 10e22 38378 . . . . . . 7 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑧𝐴   )
12 trss 4721 . . . . . . 7 (Tr 𝐴 → (𝑧𝐴𝑧𝐴))
132, 11, 12e12 38433 . . . . . 6 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑧𝐴   )
14 vex 3189 . . . . . . 7 𝑧 ∈ V
1514elpw 4136 . . . . . 6 (𝑧 ∈ 𝒫 𝐴𝑧𝐴)
1613, 15e2bir 38340 . . . . 5 (   Tr 𝐴   ,   (𝑧𝑦𝑦 ∈ 𝒫 𝐴)   ▶   𝑧 ∈ 𝒫 𝐴   )
1716in2 38312 . . . 4 (   Tr 𝐴   ▶   ((𝑧𝑦𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴)   )
1817gen12 38325 . . 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 38398 . 2 (   Tr 𝐴   ▶   Tr 𝒫 𝐴   )
2120in1 38269 1 (Tr 𝐴 → Tr 𝒫 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1478   ∈ wcel 1987   ⊆ wss 3555  𝒫 cpw 4130  Tr wtr 4712 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3188  df-in 3562  df-ss 3569  df-pw 4132  df-uni 4403  df-tr 4713  df-vd1 38268  df-vd2 38276 This theorem is referenced by: (None)
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