Theorem trsspwALT3 45993

Description: Short predicate calculus proof of the left-to-right implication of dftr4 5221. 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 45992, which is the virtual deduction proof trsspwALT 45991 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 5225	. . 3 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
2		vex 3458	. . . 4 ⊢ 𝑥 ∈ V
3	2	elpw 4564	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
4	1, 3	imbitrrdi 255	. 2 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴))
5	4	ssrdv 3919	1 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ 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-ral 3081  df-v 3456  df-ss 3901  df-pw 4562  df-uni 4871  df-tr 5216

This theorem is used by: (None)