Theorem trsspwALT 44814

Description: Virtual deduction proof of the left-to-right implication of dftr4 5224. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 5224 without accumulating results. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem trsspwALT

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ss 3934	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴))
2		idn1 44571	. . . . . . 7 ⊢ (   Tr 𝐴   ▶   Tr 𝐴   )
3		idn2 44610	. . . . . . 7 ⊢ (   Tr 𝐴   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   )
4		trss 5228	. . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
5	2, 3, 4	e12 44720	. . . . . 6 ⊢ (   Tr 𝐴   ,   𝑥 ∈ 𝐴   ▶   𝑥 ⊆ 𝐴   )
6		vex 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6	elpw 4570	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
8	5, 7	e2bir 44630	. . . . 5 ⊢ (   Tr 𝐴   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
9	8	in2 44602	. . . 4 ⊢ (   Tr 𝐴   ▶   (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)   )
10	9	gen11 44613	. . 3 ⊢ (   Tr 𝐴   ▶   ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)   )
11		biimpr 220	. . 3 ⊢ ((𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴) → 𝐴 ⊆ 𝒫 𝐴))
12	1, 10, 11	e01 44688	. 2 ⊢ (   Tr 𝐴   ▶   𝐴 ⊆ 𝒫 𝐴   )
13	12	in1 44568	1 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   ∈ wcel 2109   ⊆ wss 3917  𝒫 cpw 4566  Tr wtr 5217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-v 3452  df-ss 3934  df-pw 4568  df-uni 4875  df-tr 5218  df-vd1 44567  df-vd2 44575

This theorem is referenced by: (None)