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 Description: Virtual deduction proof of trsspwALT 38563. This proof is the same as the proof of trsspwALT 38563 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)

Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 dfss2 3576 . . 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 4726 . . . . . . 7 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
52, 3, 4sylsyld 61 . . . . . 6 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
6 vex 3192 . . . . . . 7 𝑥 ∈ V
76elpw 4141 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
85, 7syl6ibr 242 . . . . 5 (Tr 𝐴 → (𝑥𝐴𝑥 ∈ 𝒫 𝐴))
98idiALT 38200 . . . 4 (Tr 𝐴 → (𝑥𝐴𝑥 ∈ 𝒫 𝐴))
109alrimiv 1852 . . 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, 11mpsyl 68 . 2 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
1312idiALT 38200 1 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1478   ∈ wcel 1987   ⊆ wss 3559  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 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 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 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-v 3191  df-in 3566  df-ss 3573  df-pw 4137  df-uni 4408  df-tr 4718 This theorem is referenced by: (None)
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