Theorem trsspwALT 44850

Description: Virtual deduction proof of the left-to-right implication of dftr4 5199. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 5199 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 3914	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴))
2		idn1 44607	. . . . . . 7 ⊢ (   Tr 𝐴   ▶   Tr 𝐴   )
3		idn2 44646	. . . . . . 7 ⊢ (   Tr 𝐴   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   )
4		trss 5203	. . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
5	2, 3, 4	e12 44756	. . . . . 6 ⊢ (   Tr 𝐴   ,   𝑥 ∈ 𝐴   ▶   𝑥 ⊆ 𝐴   )
6		vex 3440	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6	elpw 4549	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
8	5, 7	e2bir 44666	. . . . 5 ⊢ (   Tr 𝐴   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
9	8	in2 44638	. . . 4 ⊢ (   Tr 𝐴   ▶   (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)   )
10	9	gen11 44649	. . 3 ⊢ (   Tr 𝐴   ▶   ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)   )
11		biimpr 220	. . 3 ⊢ ((𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴) → 𝐴 ⊆ 𝒫 𝐴))
12	1, 10, 11	e01 44724	. 2 ⊢ (   Tr 𝐴   ▶   𝐴 ⊆ 𝒫 𝐴   )
13	12	in1 44604	1 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1539   ∈ wcel 2111   ⊆ wss 3897  𝒫 cpw 4545  Tr wtr 5193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-v 3438  df-ss 3914  df-pw 4547  df-uni 4855  df-tr 5194  df-vd1 44603  df-vd2 44611

This theorem is referenced by: (None)