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Theorem pwtrrVD 41024
Description: Virtual deduction proof of pwtr 5342; see pwtrVD 41023 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 5171 . . 3 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
2 idn1 40773 . . . . . . . 8 (   Tr 𝒫 𝐴   ▶   Tr 𝒫 𝐴   )
3 idn2 40812 . . . . . . . . 9 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   (𝑧𝑦𝑦𝐴)   )
4 simpr 485 . . . . . . . . 9 ((𝑧𝑦𝑦𝐴) → 𝑦𝐴)
53, 4e2 40830 . . . . . . . 8 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦𝐴   )
6 pwtrrVD.1 . . . . . . . . 9 𝐴 ∈ V
76pwid 4561 . . . . . . . 8 𝐴 ∈ 𝒫 𝐴
8 trel 5176 . . . . . . . . 9 (Tr 𝒫 𝐴 → ((𝑦𝐴𝐴 ∈ 𝒫 𝐴) → 𝑦 ∈ 𝒫 𝐴))
98expd 416 . . . . . . . 8 (Tr 𝒫 𝐴 → (𝑦𝐴 → (𝐴 ∈ 𝒫 𝐴𝑦 ∈ 𝒫 𝐴)))
102, 5, 7, 9e120 40862 . . . . . . 7 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦 ∈ 𝒫 𝐴   )
11 elpwi 4554 . . . . . . 7 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
1210, 11e2 40830 . . . . . 6 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦𝐴   )
13 simpl 483 . . . . . . 7 ((𝑧𝑦𝑦𝐴) → 𝑧𝑦)
143, 13e2 40830 . . . . . 6 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑧𝑦   )
15 ssel 3965 . . . . . 6 (𝑦𝐴 → (𝑧𝑦𝑧𝐴))
1612, 14, 15e22 40870 . . . . 5 (   Tr 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑧𝐴   )
1716in2 40804 . . . 4 (   Tr 𝒫 𝐴   ▶   ((𝑧𝑦𝑦𝐴) → 𝑧𝐴)   )
1817gen12 40817 . . 3 (   Tr 𝒫 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)   )
19 biimpr 221 . . 3 ((Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)) → (∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴) → Tr 𝐴))
201, 18, 19e01 40890 . 2 (   Tr 𝒫 𝐴   ▶   Tr 𝐴   )
2120in1 40770 1 (Tr 𝒫 𝐴 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1528  wcel 2107  Vcvv 3500  wss 3940  𝒫 cpw 4542  Tr wtr 5169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-v 3502  df-in 3947  df-ss 3956  df-pw 4544  df-uni 4838  df-tr 5170  df-vd1 40769  df-vd2 40777
This theorem is referenced by: (None)
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