Theorem trsspwALT 45554

Description: Virtual deduction proof of the left-to-right implication of dftr4 5224. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 5224 without accumulating results. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem trsspwALT

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ss 3922	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴))
2		idn1 45311	. . . . . . 7 ⊢ (   Tr 𝐴   ▶   Tr 𝐴   )
3		idn2 45350	. . . . . . 7 ⊢ (   Tr 𝐴   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   )
4		trss 5228	. . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
5	2, 3, 4	e12 45460	. . . . . 6 ⊢ (   Tr 𝐴   ,   𝑥 ∈ 𝐴   ▶   𝑥 ⊆ 𝐴   )
6		vex 3459	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6	elpw 4566	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
8	5, 7	e2bir 45370	. . . . 5 ⊢ (   Tr 𝐴   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
9	8	in2 45342	. . . 4 ⊢ (   Tr 𝐴   ▶   (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)   )
10	9	gen11 45353	. . 3 ⊢ (   Tr 𝐴   ▶   ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)   )
11		biimpr 223	. . 3 ⊢ ((𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴) → 𝐴 ⊆ 𝒫 𝐴))
12	1, 10, 11	e01 45428	. 2 ⊢ (   Tr 𝐴   ▶   𝐴 ⊆ 𝒫 𝐴   )
13	12	in1 45308	1 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568   ∈ wcel 2143   ⊆ wss 3905  𝒫 cpw 4562  Tr wtr 5218

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-v 3457  df-ss 3922  df-pw 4564  df-uni 4873  df-tr 5219  df-vd1 45307  df-vd2 45315

This theorem is used by: (None)