Theorem trsspwALT3 41147

Description: Short predicate calculus proof of the left-to-right implication of dftr4 5169. 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 41146, which is the virtual deduction proof trsspwALT 41145 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 5173	. . 3 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
2		vex 3497	. . . 4 ⊢ 𝑥 ∈ V
3	2	elpw 4545	. . 3 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
4	1, 3	syl6ibr 254	. 2 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴))
5	4	ssrdv 3972	1 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴)

Syntax hints:   → wi 4   ∈ wcel 2110   ⊆ wss 3935  𝒫 cpw 4538  Tr wtr 5164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3496  df-in 3942  df-ss 3951  df-pw 4540  df-uni 4832  df-tr 5165

This theorem is referenced by: (None)