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Theorem trsspwALT 43081
Description: Virtual deduction proof of the left-to-right implication of dftr4 5229. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 5229 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 dfss2 3930 . . 3 (𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴))
2 idn1 42837 . . . . . . 7 (   Tr 𝐴   ▶   Tr 𝐴   )
3 idn2 42876 . . . . . . 7 (   Tr 𝐴   ,   𝑥𝐴   ▶   𝑥𝐴   )
4 trss 5233 . . . . . . 7 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
52, 3, 4e12 42987 . . . . . 6 (   Tr 𝐴   ,   𝑥𝐴   ▶   𝑥𝐴   )
6 vex 3449 . . . . . . 7 𝑥 ∈ V
76elpw 4564 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
85, 7e2bir 42896 . . . . 5 (   Tr 𝐴   ,   𝑥𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
98in2 42868 . . . 4 (   Tr 𝐴   ▶   (𝑥𝐴𝑥 ∈ 𝒫 𝐴)   )
109gen11 42879 . . 3 (   Tr 𝐴   ▶   𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴)   )
11 biimpr 219 . . 3 ((𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴)) → (∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴) → 𝐴 ⊆ 𝒫 𝐴))
121, 10, 11e01 42954 . 2 (   Tr 𝐴   ▶   𝐴 ⊆ 𝒫 𝐴   )
1312in1 42834 1 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1539  wcel 2106  wss 3910  𝒫 cpw 4560  Tr wtr 5222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-v 3447  df-in 3917  df-ss 3927  df-pw 4562  df-uni 4866  df-tr 5223  df-vd1 42833  df-vd2 42841
This theorem is referenced by: (None)
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