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Theorem sspwtrALT 38518
Description: Virtual deduction proof of sspwtr 38517. This proof is the same as the proof of sspwtr 38517 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 4719 . . 3 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
2 simpr 477 . . . . . 6 ((𝑧𝑦𝑦𝐴) → 𝑦𝐴)
3 ssel 3582 . . . . . 6 (𝐴 ⊆ 𝒫 𝐴 → (𝑦𝐴𝑦 ∈ 𝒫 𝐴))
4 elpwi 4145 . . . . . 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 473 . . . . . 6 ((𝑧𝑦𝑦𝐴) → 𝑧𝑦)
86, 7syl6 35 . . . . 5 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝑦))
9 ssel 3582 . . . . 5 (𝑦𝐴 → (𝑧𝑦𝑧𝐴))
105, 8, 9syl6c 70 . . . 4 (𝐴 ⊆ 𝒫 𝐴 → ((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
1110alrimivv 1858 . . 3 (𝐴 ⊆ 𝒫 𝐴 → ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
12 biimpr 210 . . 3 ((Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)) → (∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴) → Tr 𝐴))
131, 11, 12mpsyl 68 . 2 (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)
1413idiALT 38151 1 (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478  wcel 1992  wss 3560  𝒫 cpw 4135  Tr wtr 4717
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193  df-in 3567  df-ss 3574  df-pw 4137  df-uni 4408  df-tr 4718
This theorem is referenced by: (None)
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