Theorem trsspwALT 41145

Description: Virtual deduction proof of the left-to-right implication of dftr4 5169. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 5169 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 3954	. . 3 ⊢ (𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴))
2		idn1 40901	. . . . . . 7 ⊢ (   Tr 𝐴   ▶   Tr 𝐴   )
3		idn2 40940	. . . . . . 7 ⊢ (   Tr 𝐴   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝐴   )
4		trss 5173	. . . . . . 7 ⊢ (Tr 𝐴 → (𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐴))
5	2, 3, 4	e12 41051	. . . . . 6 ⊢ (   Tr 𝐴   ,   𝑥 ∈ 𝐴   ▶   𝑥 ⊆ 𝐴   )
6		vex 3497	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6	elpw 4545	. . . . . 6 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
8	5, 7	e2bir 40960	. . . . 5 ⊢ (   Tr 𝐴   ,   𝑥 ∈ 𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
9	8	in2 40932	. . . 4 ⊢ (   Tr 𝐴   ▶   (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)   )
10	9	gen11 40943	. . 3 ⊢ (   Tr 𝐴   ▶   ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)   )
11		biimpr 222	. . 3 ⊢ ((𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴)) → (∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝒫 𝐴) → 𝐴 ⊆ 𝒫 𝐴))
12	1, 10, 11	e01 41018	. 2 ⊢ (   Tr 𝐴   ▶   𝐴 ⊆ 𝒫 𝐴   )
13	12	in1 40898	1 ⊢ (Tr 𝐴 → 𝐴 ⊆ 𝒫 𝐴)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1531   ∈ wcel 2110   ⊆ wss 3935  𝒫 cpw 4538  Tr wtr 5164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-v 3496  df-in 3942  df-ss 3951  df-pw 4540  df-uni 4832  df-tr 5165  df-vd1 40897  df-vd2 40905

This theorem is referenced by: (None)