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Theorem sspwtrALT2 39580
Description: Short predicate calculus proof of the right-to-left implication of dftr4 4892. A class which is a subclass of its power class is transitive. This proof was constructed by applying Metamath's minimize command to the proof of sspwtrALT 39574, which is the virtual deduction proof sspwtr 39573 without virtual deductions. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwtrALT2 (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

Proof of Theorem sspwtrALT2
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3746 . . . . . 6 (𝐴 ⊆ 𝒫 𝐴 → (𝑦𝐴𝑦 ∈ 𝒫 𝐴))
21adantld 478 . . . . 5 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑦 ∈ 𝒫 𝐴))
3 elpwi 4308 . . . . 5 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
42, 3syl6 35 . . . 4 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑦𝐴))
5 simpl 468 . . . . 5 ((𝑧𝑦𝑦𝐴) → 𝑧𝑦)
65a1i 11 . . . 4 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝑦))
7 ssel 3746 . . . 4 (𝑦𝐴 → (𝑧𝑦𝑧𝐴))
84, 6, 7syl6c 70 . . 3 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
98alrimivv 2008 . 2 (𝐴 ⊆ 𝒫 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
10 dftr2 4889 . 2 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
119, 10sylibr 224 1 (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1629  wcel 2145  wss 3723  𝒫 cpw 4298  Tr wtr 4887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730  df-ss 3737  df-pw 4300  df-uni 4576  df-tr 4888
This theorem is referenced by: (None)
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