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Theorem sspwtr 39365
 Description: Virtual deduction proof of the right-to-left implication of dftr4 4790. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 39365 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 4787 . . 3 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
2 idn1 39107 . . . . . . . 8 (   𝐴 ⊆ 𝒫 𝐴   ▶   𝐴 ⊆ 𝒫 𝐴   )
3 idn2 39155 . . . . . . . . 9 (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   (𝑧𝑦𝑦𝐴)   )
4 simpr 476 . . . . . . . . 9 ((𝑧𝑦𝑦𝐴) → 𝑦𝐴)
53, 4e2 39173 . . . . . . . 8 (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦𝐴   )
6 ssel 3630 . . . . . . . 8 (𝐴 ⊆ 𝒫 𝐴 → (𝑦𝐴𝑦 ∈ 𝒫 𝐴))
72, 5, 6e12 39268 . . . . . . 7 (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦 ∈ 𝒫 𝐴   )
8 elpwi 4201 . . . . . . 7 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
97, 8e2 39173 . . . . . 6 (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦𝐴   )
10 simpl 472 . . . . . . 7 ((𝑧𝑦𝑦𝐴) → 𝑧𝑦)
113, 10e2 39173 . . . . . 6 (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑧𝑦   )
12 ssel 3630 . . . . . 6 (𝑦𝐴 → (𝑧𝑦𝑧𝐴))
139, 11, 12e22 39213 . . . . 5 (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑧𝐴   )
1413in2 39147 . . . 4 (   𝐴 ⊆ 𝒫 𝐴   ▶   ((𝑧𝑦𝑦𝐴) → 𝑧𝐴)   )
1514gen12 39160 . . 3 (   𝐴 ⊆ 𝒫 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)   )
16 biimpr 210 . . 3 ((Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)) → (∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴) → Tr 𝐴))
171, 15, 16e01 39233 . 2 (   𝐴 ⊆ 𝒫 𝐴   ▶   Tr 𝐴   )
1817in1 39104 1 (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383  ∀wal 1521   ∈ wcel 2030   ⊆ wss 3607  𝒫 cpw 4191  Tr wtr 4785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-in 3614  df-ss 3621  df-pw 4193  df-uni 4469  df-tr 4786  df-vd1 39103  df-vd2 39111 This theorem is referenced by: (None)
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