Theorem pwtrrVD 42445

Description: Virtual deduction proof of pwtr 5368; see pwtrVD 42444 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.

Step	Hyp	Ref	Expression
1		dftr2 5193	. . 3 ⊢ (Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
2		idn1 42194	. . . . . . . 8 ⊢ (   Tr 𝒫 𝐴   ▶   Tr 𝒫 𝐴   )
3		idn2 42233	. . . . . . . . 9 ⊢ (   Tr 𝒫 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   )
4		simpr 485	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
5	3, 4	e2 42251	. . . . . . . 8 ⊢ (   Tr 𝒫 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   ▶   𝑦 ∈ 𝐴   )
6		pwtrrVD.1	. . . . . . . . 9 ⊢ 𝐴 ∈ V
7	6	pwid 4557	. . . . . . . 8 ⊢ 𝐴 ∈ 𝒫 𝐴
8		trel 5198	. . . . . . . . 9 ⊢ (Tr 𝒫 𝐴 → ((𝑦 ∈ 𝐴 ∧ 𝐴 ∈ 𝒫 𝐴) → 𝑦 ∈ 𝒫 𝐴))
9	8	expd 416	. . . . . . . 8 ⊢ (Tr 𝒫 𝐴 → (𝑦 ∈ 𝐴 → (𝐴 ∈ 𝒫 𝐴 → 𝑦 ∈ 𝒫 𝐴)))
10	2, 5, 7, 9	e120 42283	. . . . . . 7 ⊢ (   Tr 𝒫 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   ▶   𝑦 ∈ 𝒫 𝐴   )
11		elpwi 4542	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝐴 → 𝑦 ⊆ 𝐴)
12	10, 11	e2 42251	. . . . . 6 ⊢ (   Tr 𝒫 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   ▶   𝑦 ⊆ 𝐴   )
13		simpl 483	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝑦)
14	3, 13	e2 42251	. . . . . 6 ⊢ (   Tr 𝒫 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   ▶   𝑧 ∈ 𝑦   )
15		ssel 3914	. . . . . 6 ⊢ (𝑦 ⊆ 𝐴 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴))
16	12, 14, 15	e22 42291	. . . . 5 ⊢ (   Tr 𝒫 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   ▶   𝑧 ∈ 𝐴   )
17	16	in2 42225	. . . 4 ⊢ (   Tr 𝒫 𝐴   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)   )
18	17	gen12 42238	. . 3 ⊢ (   Tr 𝒫 𝐴   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)   )
19		biimpr 219	. . 3 ⊢ ((Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) → (∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴) → Tr 𝐴))
20	1, 18, 19	e01 42311	. 2 ⊢ (   Tr 𝒫 𝐴   ▶   Tr 𝐴   )
21	20	in1 42191	1 ⊢ (Tr 𝒫 𝐴 → Tr 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   ∈ wcel 2106  Vcvv 3432   ⊆ wss 3887  𝒫 cpw 4533  Tr wtr 5191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904  df-pw 4535  df-uni 4840  df-tr 5192  df-vd1 42190  df-vd2 42198

This theorem is referenced by: (None)