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Theorem sspwtrALT 41891
 Description: Virtual deduction proof of sspwtr 41890. This proof is the same as the proof of sspwtr 41890 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A class which is a subclass of its power class is transitive. (Contributed by Alan Sare, 3-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwtrALT (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

Proof of Theorem sspwtrALT
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 5138 . . 3 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
2 simpr 489 . . . . . 6 ((𝑧𝑦𝑦𝐴) → 𝑦𝐴)
3 ssel 3886 . . . . . 6 (𝐴 ⊆ 𝒫 𝐴 → (𝑦𝐴𝑦 ∈ 𝒫 𝐴))
4 elpwi 4501 . . . . . 6 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
52, 3, 4syl56 36 . . . . 5 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑦𝐴))
6 idd 24 . . . . . 6 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → (𝑧𝑦𝑦𝐴)))
7 simpl 487 . . . . . 6 ((𝑧𝑦𝑦𝐴) → 𝑧𝑦)
86, 7syl6 35 . . . . 5 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝑦))
9 ssel 3886 . . . . 5 (𝑦𝐴 → (𝑧𝑦𝑧𝐴))
105, 8, 9syl6c 70 . . . 4 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
1110alrimivv 1930 . . 3 (𝐴 ⊆ 𝒫 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
12 biimpr 223 . . 3 ((Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)) → (∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴) → Tr 𝐴))
131, 11, 12mpsyl 68 . 2 (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)
1413idiALT 41546 1 (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1537   ∈ wcel 2112   ⊆ wss 3859  𝒫 cpw 4492  Tr wtr 5136 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-v 3412  df-in 3866  df-ss 3876  df-pw 4494  df-uni 4797  df-tr 5137 This theorem is referenced by: (None)
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