Theorem trsspwALT2 44781

Description: Virtual deduction proof of trsspwALT 44780. This proof is the same as the proof of trsspwALT 44780 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A transitive class is a subset of its power class. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
trsspwALT2	⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴)

Proof of Theorem trsspwALT2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ss 3928	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴))
2		id 22	. . . . . . 7 ⊢ (Tr 𝐴 → Tr 𝐴)
3		idd 24	. . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
4		trss 5220	. . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
5	2, 3, 4	sylsyld 61	. . . . . 6 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
6		vex 3448	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6	elpw 4563	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
8	5, 7	imbitrrdi 252	. . . . 5 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴))
9	8	idiALT 44441	. . . 4 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴))
10	9	alrimiv 1927	. . 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	mpsyl 68	. 2 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴)
13	12	idiALT 44441	1 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   ∈ wcel 2109   ⊆ wss 3911  𝒫 cpw 4559  Tr wtr 5209

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-v 3446  df-ss 3928  df-pw 4561  df-uni 4868  df-tr 5210

This theorem is referenced by: (None)