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Theorem trsspwALT2 42439
Description: Virtual deduction proof of trsspwALT 42438. This proof is the same as the proof of trsspwALT 42438 except each virtual deduction symbol is replaced by its non-virtual deduction symbol equivalent. A transitive class is a subset of its power class. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
trsspwALT2 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)

Proof of Theorem trsspwALT2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfss2 3907 . . 3 (𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴))
2 id 22 . . . . . . 7 (Tr 𝐴 → Tr 𝐴)
3 idd 24 . . . . . . 7 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
4 trss 5200 . . . . . . 7 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
52, 3, 4sylsyld 61 . . . . . 6 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
6 vex 3436 . . . . . . 7 𝑥 ∈ V
76elpw 4537 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
85, 7syl6ibr 251 . . . . 5 (Tr 𝐴 → (𝑥𝐴𝑥 ∈ 𝒫 𝐴))
98idiALT 42097 . . . 4 (Tr 𝐴 → (𝑥𝐴𝑥 ∈ 𝒫 𝐴))
109alrimiv 1930 . . 3 (Tr 𝐴 → ∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴))
11 biimpr 219 . . 3 ((𝐴 ⊆ 𝒫 𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴)) → (∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴) → 𝐴 ⊆ 𝒫 𝐴))
121, 10, 11mpsyl 68 . 2 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
1312idiALT 42097 1 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wal 1537  wcel 2106  wss 3887  𝒫 cpw 4533  Tr wtr 5191
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-11 2154  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-v 3434  df-in 3894  df-ss 3904  df-pw 4535  df-uni 4840  df-tr 5192
This theorem is referenced by: (None)
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