Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sspwtrALT2 Structured version   Visualization version   GIF version

Theorem sspwtrALT2 42481
Description: Short predicate calculus proof of the right-to-left implication of dftr4 5205. 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 42480, which is the virtual deduction proof sspwtr 42479 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 3919 . . . . . 6 (𝐴 ⊆ 𝒫 𝐴 → (𝑦𝐴𝑦 ∈ 𝒫 𝐴))
21adantld 492 . . . . 5 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑦 ∈ 𝒫 𝐴))
3 elpwi 4546 . . . . 5 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
42, 3syl6 35 . . . 4 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑦𝐴))
5 simpl 484 . . . . 5 ((𝑧𝑦𝑦𝐴) → 𝑧𝑦)
65a1i 11 . . . 4 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝑦))
7 ssel 3919 . . . 4 (𝑦𝐴 → (𝑧𝑦𝑧𝐴))
84, 6, 7syl6c 70 . . 3 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
98alrimivv 1929 . 2 (𝐴 ⊆ 𝒫 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
10 dftr2 5200 . 2 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
119, 10sylibr 233 1 (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wal 1537  wcel 2104  wss 3892  𝒫 cpw 4539  Tr wtr 5198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3439  df-in 3899  df-ss 3909  df-pw 4541  df-uni 4845  df-tr 5199
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator