Theorem trsspwALT2 45555

Description: Virtual deduction proof of trsspwALT 45554. This proof is the same as the proof of trsspwALT 45554 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 3922	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴))
2		id 23	. . . . . . 7 ⊢ (Tr 𝐴 → Tr 𝐴)
3		idd 25	. . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴))
4		trss 5228	. . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
5	2, 3, 4	sylsyld 62	. . . . . 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	imbitrrdi 255	. . . . 5 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴))
9	8	idiALT 45215	. . . 4 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴))
10	9	alrimiv 1957	. . 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	mpsyl 69	. 2 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴)
13	12	idiALT 45215	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

This theorem is used by: (None)