Theorem pwtrVD 45512

Description: Virtual deduction proof of pwtr 5435; see pwtrrVD 45513 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.

Step	Hyp	Ref	Expression
1		dftr2 5221	. . 3 ⊢ (Tr 𝒫 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴))
2		idn1 45263	. . . . . . 7 ⊢ (   Tr 𝐴   ▶   Tr 𝐴   )
3		idn2 45302	. . . . . . . . . 10 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)   )
4		simpr 489	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑦 ∈ 𝒫 𝐴)
5	3, 4	e2 45320	. . . . . . . . 9 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)   ▶   𝑦 ∈ 𝒫 𝐴   )
6		elpwi 4570	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝐴 → 𝑦 ⊆ 𝐴)
7	5, 6	e2 45320	. . . . . . . 8 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)   ▶   𝑦 ⊆ 𝐴   )
8		simpl 487	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝑦)
9	3, 8	e2 45320	. . . . . . . 8 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)   ▶   𝑧 ∈ 𝑦   )
10		ssel 3932	. . . . . . . 8 ⊢ (𝑦 ⊆ 𝐴 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴))
11	7, 9, 10	e22 45360	. . . . . . 7 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)   ▶   𝑧 ∈ 𝐴   )
12		trss 5229	. . . . . . 7 ⊢ (Tr 𝐴 → (𝑧 ∈ 𝐴 → 𝑧 ⊆ 𝐴))
13	2, 11, 12	e12 45412	. . . . . 6 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)   ▶   𝑧 ⊆ 𝐴   )
14		vex 3459	. . . . . . 7 ⊢ 𝑧 ∈ V
15	14	elpw 4567	. . . . . 6 ⊢ (𝑧 ∈ 𝒫 𝐴 ↔ 𝑧 ⊆ 𝐴)
16	13, 15	e2bir 45322	. . . . 5 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)   ▶   𝑧 ∈ 𝒫 𝐴   )
17	16	in2 45294	. . . 4 ⊢ (   Tr 𝐴   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴)   )
18	17	gen12 45307	. . 3 ⊢ (   Tr 𝐴   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴)   )
19		biimpr 223	. . 3 ⊢ ((Tr 𝒫 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴)) → (∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴) → Tr 𝒫 𝐴))
20	1, 18, 19	e01 45380	. 2 ⊢ (   Tr 𝐴   ▶   Tr 𝒫 𝐴   )
21	20	in1 45260	1 ⊢ (Tr 𝐴 → Tr 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1568   ∈ wcel 2143   ⊆ wss 3906  𝒫 cpw 4563  Tr wtr 5219

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-v 3457  df-ss 3923  df-pw 4565  df-uni 4874  df-tr 5220  df-vd1 45259  df-vd2 45267

This theorem is referenced by: (None)