Theorem trsspwALT 42327

Description: Virtual deduction proof of the left-to-right implication of dftr4 5192. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 5192 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		dfss2 3903	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴))
2		idn1 42083	. . . . . . 7 ⊢ (   Tr 𝐴   ▶   Tr 𝐴   )
3		idn2 42122	. . . . . . 7 ⊢ (   Tr 𝐴   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   )
4		trss 5196	. . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
5	2, 3, 4	e12 42233	. . . . . 6 ⊢ (   Tr 𝐴   ,   𝑥 ∈ 𝐴   ▶   𝑥 ⊆ 𝐴   )
6		vex 3426	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6	elpw 4534	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
8	5, 7	e2bir 42142	. . . . 5 ⊢ (   Tr 𝐴   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
9	8	in2 42114	. . . 4 ⊢ (   Tr 𝐴   ▶   (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)   )
10	9	gen11 42125	. . 3 ⊢ (   Tr 𝐴   ▶   ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)   )
11		biimpr 219	. . 3 ⊢ ((𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴) → 𝐴 ⊆ 𝒫 𝐴))
12	1, 10, 11	e01 42200	. 2 ⊢ (   Tr 𝐴   ▶   𝐴 ⊆ 𝒫 𝐴   )
13	12	in1 42080	1 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   ∈ 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  df-vd1 42079  df-vd2 42087

This theorem is referenced by: (None)