Theorem trsspwALT3 44791

Description: Short predicate calculus proof of the left-to-right implication of dftr4 5290. A transitive class is a subset of its power class. This proof was constructed by applying Metamath's minimize command to the proof of trsspwALT2 44790, which is the virtual deduction proof trsspwALT 44789 without virtual deductions. (Contributed by Alan Sare, 30-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
trsspwALT3	⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴)

Proof of Theorem trsspwALT3

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		trss 5294	. . 3 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
2		vex 3492	. . . 4 ⊢ 𝑥 ∈ V
3	2	elpw 4626	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
4	1, 3	imbitrrdi 252	. 2 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴))
5	4	ssrdv 4014	1 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2108   ⊆ wss 3976  𝒫 cpw 4622  Tr wtr 5283

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-v 3490  df-ss 3993  df-pw 4624  df-uni 4932  df-tr 5284

This theorem is referenced by: (None)