Theorem sspwtrALT2 42332

Description: Short predicate calculus proof of the right-to-left implication of dftr4 5192. A class which is a subclass of its power class is transitive. This proof was constructed by applying Metamath's minimize command to the proof of sspwtrALT 42331, which is the virtual deduction proof sspwtr 42330 without virtual deductions. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
sspwtrALT2	⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

Proof of Theorem sspwtrALT2

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

Step	Hyp	Ref	Expression
1		ssel 3910	. . . . . 6 ⊢ (𝐴 ⊆ 𝒫 𝐴 → (𝑦 ∈ 𝐴 → 𝑦 ∈ 𝒫 𝐴))
2	1	adantld 490	. . . . 5 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝒫 𝐴))
3		elpwi 4539	. . . . 5 ⊢ (𝑦 ∈ 𝒫 𝐴 → 𝑦 ⊆ 𝐴)
4	2, 3	syl6 35	. . . 4 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑦 ⊆ 𝐴))
5		simpl 482	. . . . 5 ⊢ ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝑦)
6	5	a1i 11	. . . 4 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝑦))
7		ssel 3910	. . . 4 ⊢ (𝑦 ⊆ 𝐴 → (𝑧 ∈ 𝑦 → 𝑧 ∈ 𝐴))
8	4, 6, 7	syl6c 70	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
9	8	alrimivv 1932	. 2 ⊢ (𝐴 ⊆ 𝒫 𝐴 → ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
10		dftr2 5189	. 2 ⊢ (Tr 𝐴 ↔ ∀𝑧∀𝑦((𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) → 𝑧 ∈ 𝐴))
11	9, 10	sylibr 233	1 ⊢ (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   ∈ wcel 2108   ⊆ wss 3883  𝒫 cpw 4530  Tr wtr 5187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ss 3900  df-pw 4532  df-uni 4837  df-tr 5188

This theorem is referenced by: (None)