Theorem sspwtr 39817

Description: Virtual deduction proof of the right-to-left implication of dftr4 4950. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 39817 without accumulating results. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sspwtr	⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

Proof of Theorem sspwtr

Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dftr2 4947	. . 3 ⊢ (Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
2		idn1 39560	. . . . . . . 8 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ▶   𝐴 ⊆ 𝒫 𝐴   )
3		idn2 39608	. . . . . . . . 9 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   )
4		simpr 478	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
5	3, 4	e2 39626	. . . . . . . 8 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   ▶   𝑦 ∈ 𝐴   )
6		ssel 3792	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝒫 𝐴 → (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝒫 𝐴))
7	2, 5, 6	e12 39720	. . . . . . 7 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   ▶   𝑦 ∈ 𝒫 𝐴   )
8		elpwi 4359	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝐴 → 𝑦 ⊆ 𝐴)
9	7, 8	e2 39626	. . . . . 6 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   ▶   𝑦 ⊆ 𝐴   )
10		simpl 475	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝑦)
11	3, 10	e2 39626	. . . . . 6 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   ▶   𝑧 ∈ 𝑦   )
12		ssel 3792	. . . . . 6 ⊢ (𝑦 ⊆ 𝐴 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴))
13	9, 11, 12	e22 39666	. . . . 5 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   ▶   𝑧 ∈ 𝐴   )
14	13	in2 39600	. . . 4 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)   )
15	14	gen12 39613	. . 3 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)   )
16		biimpr 212	. . 3 ⊢ ((Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)) → (∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴) → Tr 𝐴))
17	1, 15, 16	e01 39686	. 2 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ▶   Tr 𝐴   )
18	17	in1 39557	1 ⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385  ∀wal 1651   ∈ wcel 2157   ⊆ wss 3769  𝒫 cpw 4349  Tr wtr 4945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-v 3387  df-in 3776  df-ss 3783  df-pw 4351  df-uni 4629  df-tr 4946  df-vd1 39556  df-vd2 39564

This theorem is referenced by: (None)