Users' Mathboxes Mathbox for Alan Sare < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  trsspwALT2 Structured version   Visualization version   GIF version

Theorem trsspwALT2 41030
Description: Virtual deduction proof of trsspwALT 41029. This proof is the same as the proof of trsspwALT 41029 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 3952 . . 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 5172 . . . . . . 7 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
52, 3, 4sylsyld 61 . . . . . 6 (Tr 𝐴 → (𝑥𝐴𝑥𝐴))
6 vex 3495 . . . . . . 7 𝑥 ∈ V
76elpw 4542 . . . . . 6 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
85, 7syl6ibr 253 . . . . 5 (Tr 𝐴 → (𝑥𝐴𝑥 ∈ 𝒫 𝐴))
98idiALT 40688 . . . 4 (Tr 𝐴 → (𝑥𝐴𝑥 ∈ 𝒫 𝐴))
109alrimiv 1919 . . 3 (Tr 𝐴 → ∀𝑥(𝑥𝐴𝑥 ∈ 𝒫 𝐴))
11 biimpr 221 . . 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 40688 1 (Tr 𝐴𝐴 ⊆ 𝒫 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1526  wcel 2105  wss 3933  𝒫 cpw 4535  Tr wtr 5163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-v 3494  df-in 3940  df-ss 3949  df-pw 4537  df-uni 4831  df-tr 5164
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator