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Theorem sspwtr 41900
 Description: Virtual deduction proof of the right-to-left implication of dftr4 5143. A class which is a subclass of its power class is transitive. This proof corresponds to the virtual deduction proof of sspwtr 41900 without accumulating results. (Contributed by Alan Sare, 2-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwtr (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)

Proof of Theorem sspwtr
Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dftr2 5140 . . 3 (Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴))
2 idn1 41653 . . . . . . . 8 (   𝐴 ⊆ 𝒫 𝐴   ▶   𝐴 ⊆ 𝒫 𝐴   )
3 idn2 41692 . . . . . . . . 9 (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   (𝑧𝑦𝑦𝐴)   )
4 simpr 488 . . . . . . . . 9 ((𝑧𝑦𝑦𝐴) → 𝑦𝐴)
53, 4e2 41710 . . . . . . . 8 (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦𝐴   )
6 ssel 3885 . . . . . . . 8 (𝐴 ⊆ 𝒫 𝐴 → (𝑦𝐴𝑦 ∈ 𝒫 𝐴))
72, 5, 6e12 41803 . . . . . . 7 (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦 ∈ 𝒫 𝐴   )
8 elpwi 4503 . . . . . . 7 (𝑦 ∈ 𝒫 𝐴𝑦𝐴)
97, 8e2 41710 . . . . . 6 (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑦𝐴   )
10 simpl 486 . . . . . . 7 ((𝑧𝑦𝑦𝐴) → 𝑧𝑦)
113, 10e2 41710 . . . . . 6 (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑧𝑦   )
12 ssel 3885 . . . . . 6 (𝑦𝐴 → (𝑧𝑦𝑧𝐴))
139, 11, 12e22 41750 . . . . 5 (   𝐴 ⊆ 𝒫 𝐴   ,   (𝑧𝑦𝑦𝐴)   ▶   𝑧𝐴   )
1413in2 41684 . . . 4 (   𝐴 ⊆ 𝒫 𝐴   ▶   ((𝑧𝑦𝑦𝐴) → 𝑧𝐴)   )
1514gen12 41697 . . 3 (   𝐴 ⊆ 𝒫 𝐴   ▶   𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)   )
16 biimpr 223 . . 3 ((Tr 𝐴 ↔ ∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴)) → (∀𝑧𝑦((𝑧𝑦𝑦𝐴) → 𝑧𝐴) → Tr 𝐴))
171, 15, 16e01 41770 . 2 (   𝐴 ⊆ 𝒫 𝐴   ▶   Tr 𝐴   )
1817in1 41650 1 (𝐴 ⊆ 𝒫 𝐴 → Tr 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   ∈ wcel 2111   ⊆ wss 3858  𝒫 cpw 4494  Tr wtr 5138 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-in 3865  df-ss 3875  df-pw 4496  df-uni 4799  df-tr 5139  df-vd1 41649  df-vd2 41657 This theorem is referenced by: (None)
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