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Theorem trsspwALT 44838
Description: Virtual deduction proof of the left-to-right implication of dftr4 5266. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 5266 without accumulating results. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
trsspwALT (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)

Proof of Theorem trsspwALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-ss 3968 . . 3 (𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴))
2 idn1 44594 . . . . . . 7 (   Tr 𝐴   ▶   Tr 𝐴   )
3 idn2 44633 . . . . . . 7 (   Tr 𝐴   ,   𝑥𝐴   ▶   𝑥𝐴   )
4 trss 5270 . . . . . . 7 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
52, 3, 4e12 44744 . . . . . 6 (   Tr 𝐴   ,   𝑥𝐴   ▶   𝑥𝐴   )
6 vex 3484 . . . . . . 7 𝑥 ∈ V
76elpw 4604 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
85, 7e2bir 44653 . . . . 5 (   Tr 𝐴   ,   𝑥𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
98in2 44625 . . . 4 (   Tr 𝐴   ▶   (𝑥𝐴𝑥 ∈ 𝒫 𝐴)   )
109gen11 44636 . . 3 (   Tr 𝐴   ▶   𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴)   )
11 biimpr 220 . . 3 ((𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴)) → (∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴) → 𝐴 ⊆ 𝒫 𝐴))
121, 10, 11e01 44711 . 2 (   Tr 𝐴   ▶   𝐴 ⊆ 𝒫 𝐴   )
1312in1 44591 1 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1538  wcel 2108  wss 3951  𝒫 cpw 4600  Tr wtr 5259
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-v 3482  df-ss 3968  df-pw 4602  df-uni 4908  df-tr 5260  df-vd1 44590  df-vd2 44598
This theorem is referenced by: (None)
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