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Theorem sspwtrALT2 38878
Description: Short predicate calculus proof of the right-to-left implication of dftr4 4748. 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 38869, which is the virtual deduction proof sspwtr 38868 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 3589 . . . . . 6 (𝐴 ⊆ 𝒫 𝐴 → (𝑦𝐴𝑦 ∈ 𝒫 𝐴))
21adantld 483 . . . . 5 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑦 ∈ 𝒫 𝐴))
3 elpwi 4159 . . . . 5 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
42, 3syl6 35 . . . 4 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑦𝐴))
5 simpl 473 . . . . 5 ((𝑧𝑦𝑦𝐴) → 𝑧𝑦)
65a1i 11 . . . 4 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝑦))
7 ssel 3589 . . . 4 (𝑦𝐴 → (𝑧𝑦𝑧𝐴))
84, 6, 7syl6c 70 . . 3 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
98alrimivv 1854 . 2 (𝐴 ⊆ 𝒫 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
10 dftr2 4745 . 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 384  wal 1479  wcel 1988  wss 3567  𝒫 cpw 4149  Tr wtr 4743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-in 3574  df-ss 3581  df-pw 4151  df-uni 4428  df-tr 4744
This theorem is referenced by: (None)
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