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Theorem pwtrrVD 39374
Description: Virtual deduction proof of pwtr 4951; see pwtrVD 39373 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 4787 . . 3 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
2 idn1 39107 . . . . . . . 8 (   Tr 𝒫 𝐴   ▶   Tr 𝒫 𝐴   )
3 idn2 39155 . . . . . . . . 9 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   (𝑧𝑦𝑦𝐴)   )
4 simpr 476 . . . . . . . . 9 ((𝑧𝑦𝑦𝐴) → 𝑦𝐴)
53, 4e2 39173 . . . . . . . 8 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦𝐴   )
6 pwtrrVD.1 . . . . . . . . 9 𝐴 ∈ V
76pwid 4207 . . . . . . . 8 𝐴 ∈ 𝒫 𝐴
8 trel 4792 . . . . . . . . 9 (Tr 𝒫 𝐴 → ((𝑦𝐴𝐴 ∈ 𝒫 𝐴) → 𝑦 ∈ 𝒫 𝐴))
98expd 451 . . . . . . . 8 (Tr 𝒫 𝐴 → (𝑦𝐴 → (𝐴 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴)))
102, 5, 7, 9e120 39205 . . . . . . 7 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦 ∈ 𝒫 𝐴   )
11 elpwi 4201 . . . . . . 7 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
1210, 11e2 39173 . . . . . 6 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦𝐴   )
13 simpl 472 . . . . . . 7 ((𝑧𝑦𝑦𝐴) → 𝑧𝑦)
143, 13e2 39173 . . . . . 6 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑧𝑦   )
15 ssel 3630 . . . . . 6 (𝑦𝐴 → (𝑧𝑦𝑧𝐴))
1612, 14, 15e22 39213 . . . . 5 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑧𝐴   )
1716in2 39147 . . . 4 (   Tr 𝒫 𝐴   ▶   ((𝑧𝑦𝑦𝐴) → 𝑧𝐴)   )
1817gen12 39160 . . 3 (   Tr 𝒫 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)   )
19 biimpr 210 . . 3 ((Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)) → (∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴) → Tr 𝐴))
201, 18, 19e01 39233 . 2 (   Tr 𝒫 𝐴   ▶   Tr 𝐴   )
2120in1 39104 1 (Tr 𝒫 𝐴 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521  wcel 2030  Vcvv 3231  wss 3607  𝒫 cpw 4191  Tr wtr 4785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-in 3614  df-ss 3621  df-pw 4193  df-uni 4469  df-tr 4786  df-vd1 39103  df-vd2 39111
This theorem is referenced by: (None)
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