Theorem sspwtr 45992

Description: Virtual deduction proof of the right-to-left implication of dftr4 5221. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 45992 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 5217	. . 3 ⊢ (Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
2		idn1 45746	. . . . . . . 8 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ▶   𝐴 ⊆ 𝒫 𝐴   )
3		idn2 45785	. . . . . . . . 9 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   ▶   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   )
4		simpr 490	. . . . . . . . 9 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐴)
5	3, 4	e2 45803	. . . . . . . 8 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   ▶   𝑦 ∈ 𝐴   )
6		ssel 3907	. . . . . . . 8 ⊢ (𝐴 ⊆ 𝒫 𝐴 → (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝒫 𝐴))
7	2, 5, 6	e12 45895	. . . . . . 7 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   ▶   𝑦 ∈ 𝒫 𝐴   )
8		elpwi 4567	. . . . . . 7 ⊢ (𝑦 ∈ 𝒫 𝐴 → 𝑦 ⊆ 𝐴)
9	7, 8	e2 45803	. . . . . 6 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   ▶   𝑦 ⊆ 𝐴   )
10		simpl 488	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝑦)
11	3, 10	e2 45803	. . . . . 6 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   ▶   𝑧 ∈ 𝑦   )
12		ssel 3907	. . . . . 6 ⊢ (𝑦 ⊆ 𝐴 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴))
13	9, 11, 12	e22 45843	. . . . 5 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)   ▶   𝑧 ∈ 𝐴   )
14	13	in2 45777	. . . 4 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ▶   ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)   )
15	14	gen12 45790	. . 3 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ▶   ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴)   )
16		biimpr 223	. . 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 45863	. 2 ⊢ (   𝐴 ⊆ 𝒫 𝐴   ▶   Tr 𝐴   )
18	17	in1 45743	1 ⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   ∈ wcel 2147   ⊆ wss 3900  𝒫 cpw 4560  Tr wtr 5215

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-pw 4562  df-uni 4871  df-tr 5216  df-vd1 45742  df-vd2 45750

This theorem is used by: (None)