Theorem pwtrVD 41156

Description: Virtual deduction proof of pwtr 5344; see pwtrrVD 41157 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 5173	. . 3 ⊢ (Tr 𝒫 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴))
2		idn1 40906	. . . . . . 7 ⊢ (   Tr 𝐴   ▶   Tr 𝐴   )
3		idn2 40945	. . . . . . . . . 10 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)   )
4		simpr 487	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑦 ∈ 𝒫 𝐴)
5	3, 4	e2 40963	. . . . . . . . 9 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)   ▶   𝑦 ∈ 𝒫 𝐴   )
6		elpwi 4547	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝒫 𝐴 → 𝑦 ⊆ 𝐴)
7	5, 6	e2 40963	. . . . . . . 8 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)   ▶   𝑦 ⊆ 𝐴   )
8		simpl 485	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝑦)
9	3, 8	e2 40963	. . . . . . . 8 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)   ▶   𝑧 ∈ 𝑦   )
10		ssel 3960	. . . . . . . 8 ⊢ (𝑦 ⊆ 𝐴 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴))
11	7, 9, 10	e22 41003	. . . . . . 7 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)   ▶   𝑧 ∈ 𝐴   )
12		trss 5180	. . . . . . 7 ⊢ (Tr 𝐴 → (𝑧 ∈ 𝐴 → 𝑧 ⊆ 𝐴))
13	2, 11, 12	e12 41056	. . . . . 6 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)   ▶   𝑧 ⊆ 𝐴   )
14		vex 3497	. . . . . . 7 ⊢ 𝑧 ∈ V
15	14	elpw 4542	. . . . . 6 ⊢ (𝑧 ∈ 𝒫 𝐴 ↔ 𝑧 ⊆ 𝐴)
16	13, 15	e2bir 40965	. . . . 5 ⊢ (   Tr 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴)   ▶   𝑧 ∈ 𝒫 𝐴   )
17	16	in2 40937	. . . 4 ⊢ (   Tr 𝐴   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴)   )
18	17	gen12 40950	. . 3 ⊢ (   Tr 𝐴   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝒫 𝐴) → 𝑧 ∈ 𝒫 𝐴)   )
19		biimpr 222	. . 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 41023	. 2 ⊢ (   Tr 𝐴   ▶   Tr 𝒫 𝐴   )
21	20	in1 40903	1 ⊢ (Tr 𝐴 → Tr 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  ∀wal 1531   ∈ wcel 2110   ⊆ wss 3935  𝒫 cpw 4538  Tr wtr 5171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3496  df-in 3942  df-ss 3951  df-pw 4540  df-uni 4838  df-tr 5172  df-vd1 40902  df-vd2 40910

This theorem is referenced by: (None)