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Theorem sspwtrALT2 44848
Description: Short predicate calculus proof of the right-to-left implication of dftr4 5265. 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 44847, which is the virtual deduction proof sspwtr 44846 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 3976 . . . . . 6 (𝐴 ⊆ 𝒫 𝐴 → (𝑦𝐴𝑦 ∈ 𝒫 𝐴))
21adantld 490 . . . . 5 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑦 ∈ 𝒫 𝐴))
3 elpwi 4606 . . . . 5 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
42, 3syl6 35 . . . 4 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑦𝐴))
5 simpl 482 . . . . 5 ((𝑧𝑦𝑦𝐴) → 𝑧𝑦)
65a1i 11 . . . 4 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝑦))
7 ssel 3976 . . . 4 (𝑦𝐴 → (𝑧𝑦𝑧𝐴))
84, 6, 7syl6c 70 . . 3 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
98alrimivv 1927 . 2 (𝐴 ⊆ 𝒫 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
10 dftr2 5260 . 2 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
119, 10sylibr 234 1 (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1537  wcel 2107  wss 3950  𝒫 cpw 4599  Tr wtr 5258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-ss 3967  df-pw 4601  df-uni 4907  df-tr 5259
This theorem is referenced by: (None)
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