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 Description: Virtual deduction proof of the left-to-right implication of dftr4 4907. A transitive class is a subset of its power class. This proof corresponds to the virtual deduction proof of dftr4 4907 without accumulating results. (Contributed by Alan Sare, 29-Apr-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
trsspwALT (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)

Proof of Theorem trsspwALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfss2 3730 . . 3 (𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴))
2 idn1 39290 . . . . . . 7 (   Tr 𝐴   ▶   Tr 𝐴   )
3 idn2 39338 . . . . . . 7 (   Tr 𝐴   ,   𝑥𝐴   ▶   𝑥𝐴   )
4 trss 4911 . . . . . . 7 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
52, 3, 4e12 39451 . . . . . 6 (   Tr 𝐴   ,   𝑥𝐴   ▶   𝑥𝐴   )
6 vex 3341 . . . . . . 7 𝑥 ∈ V
76elpw 4306 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
85, 7e2bir 39358 . . . . 5 (   Tr 𝐴   ,   𝑥𝐴   ▶   𝑥 ∈ 𝒫 𝐴   )
98in2 39330 . . . 4 (   Tr 𝐴   ▶   (𝑥𝐴𝑥 ∈ 𝒫 𝐴)   )
109gen11 39341 . . 3 (   Tr 𝐴   ▶   𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴)   )
11 biimpr 210 . . 3 ((𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴)) → (∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴) → 𝐴 ⊆ 𝒫 𝐴))
121, 10, 11e01 39416 . 2 (   Tr 𝐴   ▶   𝐴 ⊆ 𝒫 𝐴   )
1312in1 39287 1 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1628   ∈ wcel 2137   ⊆ wss 3713  𝒫 cpw 4300  Tr wtr 4902 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-v 3340  df-in 3720  df-ss 3727  df-pw 4302  df-uni 4587  df-tr 4903  df-vd1 39286  df-vd2 39294 This theorem is referenced by: (None)
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