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Theorem trsspwALT 41517
 Description: Virtual deduction proof of the left-to-right implication of dftr4 5144. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 5144 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 3904 . . 3 (𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴))
2 idn1 41273 . . . . . . 7 (   Tr 𝐴   ▶   Tr 𝐴   )
3 idn2 41312 . . . . . . 7 (   Tr 𝐴   ,   𝑥𝐴   ▶   𝑥𝐴   )
4 trss 5148 . . . . . . 7 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
52, 3, 4e12 41423 . . . . . 6 (   Tr 𝐴   ,   𝑥𝐴   ▶   𝑥𝐴   )
6 vex 3447 . . . . . . 7 𝑥 ∈ V
76elpw 4504 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
85, 7e2bir 41332 . . . . 5 (   Tr 𝐴   ,   𝑥𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
98in2 41304 . . . 4 (   Tr 𝐴   ▶   (𝑥𝐴𝑥 ∈ 𝒫 𝐴)   )
109gen11 41315 . . 3 (   Tr 𝐴   ▶   𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴)   )
11 biimpr 223 . . 3 ((𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴)) → (∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴) → 𝐴 ⊆ 𝒫 𝐴))
121, 10, 11e01 41390 . 2 (   Tr 𝐴   ▶   𝐴 ⊆ 𝒫 𝐴   )
1312in1 41270 1 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536   ∈ wcel 2112   ⊆ wss 3884  𝒫 cpw 4500  Tr wtr 5139 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-11 2159  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-ral 3114  df-v 3446  df-in 3891  df-ss 3901  df-pw 4502  df-uni 4804  df-tr 5140  df-vd1 41269  df-vd2 41277 This theorem is referenced by: (None)
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