Theorem trsspwALT3 42329

Description: Short predicate calculus proof of the left-to-right implication of dftr4 5192. 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 42328, which is the virtual deduction proof trsspwALT 42327 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 5196	. . 3 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
2		vex 3426	. . . 4 ⊢ 𝑥 ∈ V
3	2	elpw 4534	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
4	1, 3	syl6ibr 251	. 2 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴))
5	4	ssrdv 3923	1 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴)

Syntax hints:   → wi 4   ∈ 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-11 2156  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-ral 3068  df-v 3424  df-in 3890  df-ss 3900  df-pw 4532  df-uni 4837  df-tr 5188

This theorem is referenced by: (None)