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Theorem sspwtr 44819
Description: Virtual deduction proof of the right-to-left implication of dftr4 5272. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 44819 without accumulating results. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwtr (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

Proof of Theorem sspwtr
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 5267 . . 3 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
2 idn1 44572 . . . . . . . 8 (   𝐴 ⊆ 𝒫 𝐴   ▶   𝐴 ⊆ 𝒫 𝐴   )
3 idn2 44611 . . . . . . . . 9 (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   (𝑧𝑦𝑦𝐴)   )
4 simpr 484 . . . . . . . . 9 ((𝑧𝑦𝑦𝐴) → 𝑦𝐴)
53, 4e2 44629 . . . . . . . 8 (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦𝐴   )
6 ssel 3989 . . . . . . . 8 (𝐴 ⊆ 𝒫 𝐴 → (𝑦𝐴𝑦 ∈ 𝒫 𝐴))
72, 5, 6e12 44722 . . . . . . 7 (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦 ∈ 𝒫 𝐴   )
8 elpwi 4612 . . . . . . 7 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
97, 8e2 44629 . . . . . 6 (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦𝐴   )
10 simpl 482 . . . . . . 7 ((𝑧𝑦𝑦𝐴) → 𝑧𝑦)
113, 10e2 44629 . . . . . 6 (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑧𝑦   )
12 ssel 3989 . . . . . 6 (𝑦𝐴 → (𝑧𝑦𝑧𝐴))
139, 11, 12e22 44669 . . . . 5 (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑧𝐴   )
1413in2 44603 . . . 4 (   𝐴 ⊆ 𝒫 𝐴   ▶   ((𝑧𝑦𝑦𝐴) → 𝑧𝐴)   )
1514gen12 44616 . . 3 (   𝐴 ⊆ 𝒫 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)   )
16 biimpr 220 . . 3 ((Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)) → (∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴) → Tr 𝐴))
171, 15, 16e01 44689 . 2 (   𝐴 ⊆ 𝒫 𝐴   ▶   Tr 𝐴   )
1817in1 44569 1 (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535  wcel 2106  wss 3963  𝒫 cpw 4605  Tr wtr 5265
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-ss 3980  df-pw 4607  df-uni 4913  df-tr 5266  df-vd1 44568  df-vd2 44576
This theorem is referenced by: (None)
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