Theorem trsspwALT3 45650

Description: Short predicate calculus proof of the left-to-right implication of dftr4 5222. 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 45649, which is the virtual deduction proof trsspwALT 45648 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 5226	. . 3 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
2		vex 3457	. . . 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 3940	1 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145   ⊆ wss 3902  𝒫 cpw 4560  Tr wtr 5216

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-v 3455  df-ss 3919  df-pw 4562  df-uni 4871  df-tr 5217

This theorem is used by: (None)